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href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Counting Principles</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/algebra/counting-principles/main.jpg" style="float: left; width: 100px; height: 100px;"> This page deals with all the counting principles in the HL course: Arrangements, Permutations and Combinations. You need to be familiar with combinations for the Binomial Theorem, but it is also useful in its own right to be able to work out, for example, how many ways there are of getting a hand of blackjack in a card game or finding how many different tickets are possible in a lottery. There is always more than one way of solving a problem. This page will enable to be confident in several methods, in order that you can have a flexible problem-solving strategy.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to count</p> <ul> <li>Arrangements</li> <li>Permutations</li> <li>Combinations</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/counting-principles/revision-notes_counting_principles.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/counting-principles/revision-notes_counting_principles.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Try this quiz to practise questions about arrangements</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#29aa9f07"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="29aa9f07"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Arrangements <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-434-673" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/4-bowls-small.jpg" style="width: 400px; height: 300px;"></p><p>4 different coloured bowls are arranged in a line. How many <strong>different</strong> arrangements are there?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="24"> <span class="review"></span></p></div><div class="q-explanation"><p>The objects are different, so there are 4! different arrangements</p><p>4! = 4 x 3 x 2 x 1 = 24</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/cards-1.jpg" style="width: 300px; height: 400px;"></p><p>5 cards are arranged in a line. How many <strong>different </strong>possible arrangements are there?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p>The objects are different, so there are 5! different arrangements</p><p>5! = 5 x 4 x 3 x 2 x 1 = 120</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/pirate-cards.jpg" style="width: 400px; height: 300px;"></p><p>The 6 cards are placed in a line. How many <strong>different </strong>arrangements are there?</p><p style="text-align: right;">*Note that the cards with a coin are identical</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p>There are 6 objects and 3 of them are repeated.</p><p>There are <span class="math-tex">\(\frac{6!}{3!}\)</span>arrangements</p><p><span class="math-tex">\(\frac{6!}{3!}=\frac{6\cdot5\cdot4\cdot3\cdot 2\cdot1}{3\cdot2\cdot 1}=6\cdot5\cdot4=120\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/coloured-tiles.jpg" style="width: 400px; height: 300px;"></p><p>The coloured blocks are arranged in a line.</p><p>How many <strong>different </strong>arrangements are there?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="180"> <span class="review"></span></p></div><div class="q-explanation"><p>There are 6 objects. The trapezia are repeated 2 times. The rhombus ar repeated 2 times.</p><p>There are <span class="math-tex">\(\frac{6!}{2!\cdot2!}=180\)</span> arrangements</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The letters from the word<strong> MATHEMATICS</strong> are placed in a line.</p><p>How many <strong>different</strong> arrangements are there?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{11!}{2!\cdot2!\cdot2!}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2!\cdot2!\cdot2!\cdot5!\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(11!\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{11!}{6!}\)</span></label></p></div><div class="q-explanation"><p>The are 11 objects. The <strong>M </strong>, <strong>A</strong> ,<strong>T</strong> are repeated 2 times each.</p><p>There are <span class="math-tex">\(\frac{11!}{2!\cdot2!\cdot2!}\)</span> arrangements</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/rods.jpg" style="width: 400px; height: 300px;"></p><p>The coloured rods are placed in a line. How many <strong>different</strong> arrangements are there?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\({3!\cdot2!\cdot2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(7!\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{7!}{3!\cdot2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{7!}{5!}\)</span></span></label> </p></div><div class="q-explanation"><p>There are 7 objects.The purple rods are repeated 3 times. The green rods are repeated 2 times.</p><p>There are <span class="math-tex">\(\frac{7!}{3!\cdot2!}\)</span> arrangements</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/more-rods.jpg" style="width: 400px; height: 300px;"></p><p>8 different coloured rods are placed in a line.</p><p>How many <strong>different </strong>arrangements are there so that the blue and the green rods are <strong>together</strong>?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(2\cdot7!\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{7!}{2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{8!}{2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(6!\)</span></span></label> </p></div><div class="q-explanation"><p>Think about the blue and green rods as one object.</p><p>This means that there now are 7 objects.</p><p>These 7 objects can be arranged in 7! ways.</p><p>For each of these arrangements, it is possible to reverse the order of the blue and green rods</p><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/rods2-ans.png" style="width: 400px; height: 107px;"></p><p>There are 2x7! different arrangements</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/10-cards.jpg" style="width: 300px; height: 400px;"></p><p>10 cards are placed in a line. How many <strong>different </strong>ways can they be arranged so that the cards 1, 2 & 3 are <strong>together</strong>?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(3\cdot8!\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{8!}{3!}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(7!\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(3!\cdot8!\)</span></label></p></div><div class="q-explanation"><p>Think of the cards 1, 2 & 3 as one object.</p><p>This makes 8 objects with 8! arrangements.</p><p>There are 3! ways of arranging the numbers 1, 2 & 3</p><p>For each of these arrangements, it is possible to rearrange the items 1 , 2 & 3</p><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/10-cards-ans.png" style="width: 400px; height: 184px;"></p><p>There are 3!x8!</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/6-bowls.jpg" style="width: 400px; height: 300px;"></p><p>6 different coloured bowls are placed in a line. How many arrangements are there in which the pink and red bowls are <strong>separated</strong>?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="480"> <span class="review"></span></p></div><div class="q-explanation"><p>The pink and red bowls are either <strong>separated </strong>or <strong>together</strong>.</p><p>Find the number of arrangements in total = 6!</p><p>Find the number of arrangements in which the pink and red bowls are <strong>together</strong> = 2x5!</p><p>The number of arrangements in which the pink and red bowls are <strong>separated</strong> = 6 - 2x5! = 480</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Three men and two women stand together in a line.</p><p>In how many arrangements will all the men stand next to each and all the women stand next to each other?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="24"> <span class="review"></span></p></div><div class="q-explanation"><p>Think of the men as 1 object. Think of the women as 1 object.</p><p>There are 2 ways of arranging these 2 objects</p><p>There are 3! arrangements of men. There are 2! arrangements of women.</p><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/men-women.png" style="width: 250px; height: 165px;"></p><p>There are 3! x 2! x 2 = 24 arrangements</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Try this quiz to practise questions about combinations and permutations</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#6548bc1c"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="6548bc1c"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Combinations and Permutations <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-435-673" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/dobble.jpg" style="width: 400px; height: 533px;"></p><p>In a card game, two images are chosen from the card.</p><p>How many different possible ways are there if order is not important?</p></div><div class="q-answer"><p>Number of ways = <input type="text" style="height: auto;" data-c="28"> <span class="review"></span></p></div><div class="q-explanation"><p>Since order is not important, then this is a combination. We chose 2 items from 8 objects</p><p><span class="math-tex">\(^{8}C_2=\frac{8!}{6!2!}=\frac{8\cdot7}{2}=28\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(^{n}C_2=36\)</span></p><p>Work out <strong><em>n</em></strong></p></div><div class="q-answer"><p><strong><em>n</em></strong> = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(^{n}C_2=\frac{n!}{(n-2)!2!}=\frac{n(n-1)}{2}\)</span></p><p>We need to solve <span class="math-tex">\(\frac{n(n-1)}{2}=36\)</span></p><p>n(n - 1) = 72</p><p>We can solve this quadratic equation, but it is clear that n = 9</p><p>Since 9x8 = 72</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/10-cards.jpg" style="width: 400px; height: 300px;"></p><p>These 10 cards are shuffled thoroughly and 3 are dealt.</p><p>What is the probability of getting 3 cards containing 1 , 2 & 3</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{3}{10}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{3!}{10!}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{1}{120}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{720}\)</span></span></label> </p></div><div class="q-explanation"><p>Since order is not important, then this is a combination. We chose 3 items from 10 objects</p><p><span class="math-tex">\(^{10}C_3=\frac{10!}{7!3!}=120\)</span></p><p>There is only one correct combination, so the probability is <span class="math-tex">\(\frac{1}{120}\)</span></p><hr class="hidden-separator"><p>Alternatively, we can think of the probability of getting three consecutive correct cards.</p><p>There is a <span class="math-tex">\(\frac{3}{10}\)</span> chance that the first card is correct</p><p>There is a <span class="math-tex">\(\frac{2}{9}\)</span> chance that the second card is correct</p><p>There is a <span class="math-tex">\(\frac{1}{8}\)</span> chance that the third card is correct</p><p>The probability = <span class="math-tex">\(\frac{3}{10}\cdot\frac{2}{9}\cdot\frac{1}{8}=\frac{1}{120}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/cards-to-6.jpg" style="width: 400px; height: 300px;"></p><p>These 5 cards are shuffled thoroughly and 2 cards are dealt.</p><p>What is the probability that the 1 card is dealt followed by the 2 card.</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{1}{20}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{10}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{120}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{2}{5}\)</span></span></label> </p></div><div class="q-explanation"><p>Since order is important this is a permutation question.</p><p>We chose 2 items from 5 objects</p><p><span class="math-tex">\(^{5}P_2=\frac{5!}{3!}=20\)</span></p><p>There is only one correct permutation, so the probability is <span class="math-tex">\(\frac{1}{20}\)</span></p><hr class="hidden-separator"><p>Alternatively, we can think of the probability of getting two consecutive correct cards.</p><p>There is a <span class="math-tex">\(\frac{1}{5}\)</span> chance that the first card is correct</p><p>There is a <span class="math-tex">\(\frac{2}{4}\)</span> chance that the second card is correct</p><p>The probability = <span class="math-tex">\(\frac{1}{5}\cdot\frac{1}{4}=\frac{1}{20}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/quiz-hero.jpg" style="width: 400px; height: 300px;"></p><p>Four letters are selected at random from these letters.</p><p>What is the probability that the letters can be put together to spell ZERO?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{1680}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{24}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{2}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{1}{70}\)</span></span></label> </p></div><div class="q-explanation"><p>Since order is not important, then this is a combination. We chose 4 items from 8 objects</p><p><span class="math-tex">\(^{8}C_4=\frac{8!}{4!4!}=70\)</span></p><p>There is only one correct combination, so the probability is <span class="math-tex">\(\frac{1}{70}\)</span></p><hr class="hidden-separator"><p>Alternatively, we can think of the probability of getting four consecutive correct letters.</p> <p>The probability = <span class="math-tex">\(\frac{4}{8}\cdot\frac{3}{7}\cdot\frac{2}{6}\cdot\frac{1}{5}=\frac{1}{70}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>In the game of Blackjack, you are dealt 2 cards from a pack of 52 cards. To get a score of 21, you need an ace and a card of value 10 (10, J, Q or K).</p><p>How many different ways are there of getting a score of 21?</p><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/cards.png" style="width: 600px; height: 247px;"></p></div><div class="q-answer"><p>Number of ways = <input type="text" style="height: auto;" data-c="64"> <span class="review"></span></p></div><div class="q-explanation"><p>Since order is not important, then this is a combination.</p><p>We need an ace: we chose 1 item from 4 objects</p><p>We need an 10 value card: we chose 1 item from 16 objects</p><p>There are 4x16 ways of getting 21</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/randomly.jpg" style="width: 400px; height: 300px;"></p><p>Four letters are chosen at random from RANDOMLY.<br>The probability that all letters chosen are <a href="http://en.wikipedia.org/wiki/Consonant#Letters" target="_blank">consonants</a> = <span class="math-tex">\(\frac{a}{70}\)</span></p><p>Work out the value of <strong><em>a</em></strong></p><p style="text-align: right;"><em>*For the purposes of this question, Y is not considered a consonant</em></p></div><div class="q-answer"><p><strong><em>a </em></strong>= <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p></div><div class="q-explanation"><p>Since order is not important, then this is a combination.</p><p>First find the total number of combinations of 4 letters chosen from RANDOMLY.</p><p>We chose 4 items from 8 objects: <span class="math-tex">\(^{8}C_4=\frac{8!}{4!4!}=70\)</span></p><p>Now find the number of combinations of 4 letters chosen from the consonants RNDMLY</p><p>We chose 4 items from 6 objects: <span class="math-tex">\(^{6}C_4=\frac{6!}{4!2!}=15\)</span></p><p>The probability that all letters chosen are consonants = <span class="math-tex">\(\frac{15}{70}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/combinations-and-permutations-quiz/barcelona.jpg" style="width: 300px; height: 300px;"></p><p>The Barcelona soccer team includes 1 goalkeeper, 4 defenders, 3 midfielders and 3 attackers.</p><p>The manager has a squad of 22 players to choose from: 3 goalkeepers, 8 defenders, 6 midfielders and 5 attackers.</p><p>In how many ways can the Barcelona manager pick the team of 11 players?</p></div><div class="q-answer"><p>Number of different teams = <input type="text" style="height: auto;" data-c="42000"> <span class="review"></span></p></div><div class="q-explanation"><p>We choose 1 item from 3: <span class="math-tex">\(^{3}C_1=3\)</span></p><p>We choose 4 items from 8: <span class="math-tex">\(^{8}C_4=70\)</span></p><p>We choose 3 items from 6: <span class="math-tex">\(^{6}C_3=20\)</span></p><p>We choose 3 items from 5: <span class="math-tex">\(^{5}C_3=10\)</span></p><p>There are 3 x 70 x 20 x 10 = 42 000 ways</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A history exam has 2 sections. Section A has 5 questions and section B has 3 questions.</p><p>A student must answer 3 questions.</p><p>How many different ways are there of selecting 3 questions if a student must answer at least one question from each section?</p></div><div class="q-answer"><p>Number of ways = <input type="text" style="height: auto;" data-c="45"> <span class="review"></span></p></div><div class="q-explanation"><p>The student could either answer:</p><p>2 section A questions and 1 section B question</p><p>= <span class="math-tex">\(^{5}C_2\cdot^{3}C_1=10\times3=30\)</span></p><p>or</p><p>1 section A question and 2 section B questions</p><p>= <span class="math-tex">\(^{5}C_1\cdot^{3}C_2=5\times3=15\)</span></p><p>There are 30 + 15 = 45 ways in total</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>From a class of 8 boys and 8 girls, a group of 6 students is chosen for a quiz team. The team must have an even number of boys and girls. Donald has narcissistic tendencies and thinks he knows more than everyone else.</p><p>In how many ways can the team be selected if Donald (a boy) must be in the team?</p></div><div class="q-answer"><p>Number of teams = <input type="text" style="height: auto;" data-c="1176"> <span class="review"></span></p></div><div class="q-explanation"><p>There must be 3 girls:</p><p>We choose 3 items from 8: <span class="math-tex">\(^{8}C_3=56\)</span></p><p>There must be Donald:</p><p>There is only one Donald</p><p>There must be 2 more boys</p><p>We choose 2 items from 7: <span class="math-tex">\(^{7}C_2=21\)</span></p><hr class="hidden-separator">There are 56 x 1 x 21 = 1176</div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="490"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>A team of five players is chosen from six males and 5 females.</p> <ol style="list-style-type:lower-alpha;"> <li>Determine how many different teams can be formed.</li> <li>Determine how many different teams can be formed consisting of 3 males and 2 females.</li> <li>Determine how many different teams can be formed if the team consists of more females than males</li> </ol> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Since order is not important, these are combinations</p> <p>c. The team of 5 needs to contain more females than males.</p> <p>It could have:</p> <p>3 females and 2 males</p> <p><strong>Or</strong></p> <p>4 females and 1 male</p> <p><strong>Or</strong></p> <p>5 females and 0 males</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/counting-principles/ppq_counting_principles1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/counting-principles/ppq_counting_principles1.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="491"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>A five-digit number is formed by using the digits 1-5 exactly once.</p> <ol style="list-style-type:lower-alpha;"> <li>How many five-digit numbers are there?</li> <li>How many of these five-digit numbers are even?</li> </ol> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>For the digits 1-5, the number must end in 2 or 4.</p> <p>How many arrangements end with 2?</p> <p>How many arrangements end with 4?</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/counting-principles/ppq_counting_principles2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/counting-principles/ppq_counting_principles2.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="492"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Seven students are placed at random in a line.</p> <ol style="list-style-type:lower-alpha;"> <li>How many different arrangements are there?</li> <li>What is the probability that the two youngest students are separated?</li> </ol> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Treat the two youngest students as one object.</p> <p>Consider finding the probability that the two youngest students are together.<content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs 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