A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is __________
Answer (integer)
546
Solution
<p>Among 12 courses, 5 courses are of language.</p>
<p>$\therefore$ Remaining 7 are different courses.</p>
<p>Now, number of ways to select 5 courses where at most 2 language courses present.</p>
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<tr>
<th class="tg-7btt"></th>
<th class="tg-7btt">Language</th>
<th class="tg-7btt">Different</th>
<th class="tg-7btt">Number of ways</th>
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<td class="tg-c3ow">Case 1</td>
<td class="tg-c3ow">0</td>
<td class="tg-c3ow">5</td>
<td class="tg-c3ow">${}^5{C_0} \times {}^7{C_5}$</td>
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<td class="tg-c3ow">Case 2</td>
<td class="tg-c3ow">1</td>
<td class="tg-c3ow">4</td>
<td class="tg-c3ow">${}^5{C_1} \times {}^7{C_4}$</td>
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<tr>
<td class="tg-c3ow">Case 3</td>
<td class="tg-c3ow">2</td>
<td class="tg-c3ow">3</td>
<td class="tg-c3ow">${}^5{C_2} \times {}^7{C_3}$</td>
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<p>$\therefore$ Total number of ways</p>
<p>$$ = {}^5{C_0} \times {}^7{C_5} + {}^5{C_1} \times {}^7{C_4} + {}^5{C_2} \times {}^7{C_3}$$</p>
<p>$= 546$</p>
About this question
Subject: Mathematics · Chapter: Permutations and Combinations · Topic: Fundamental Counting Principle
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