Permutations & Combinations
Grade 11
Question:
<p>A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:</p>
<p style="display:inline">560</p>
<p style="display:inline">575</p>
<p style="display:inline">1050</p>
<p style="display:inline">1625</p>
Step-by-Step Solution
Key Concept: Identify all distinct cases that satisfy the given population constraints and use combinations to sum the selection ways for each valid case.
<table border="1" cellpadding="3" cellspacing="0" style="width:100%">
<thead>
<tr>
<th scope="col">Indians</th>
<th scope="col">Foreigners</th>
<th scope="col">Number of Ways</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">2</td>
<td style="text-align:center">4</td>
<td style="text-align:center"><span class="math-tex">\({ }^6 C _2 \times{ }^8 C _4\)</span> = 1050</td>
</tr>
<tr>
<td style="text-align:center">3</td>
<td style="text-align:center">6</td>
<td style="text-align:center"><span class="math-tex">\({ }^6 C _3 \times{ }^8 C _6\)</span> = 560</td>
</tr>
<tr>
<td style="text-align:center">4</td>
<td style="text-align:center">8</td>
<td style="text-align:center"><span class="math-tex">\({ }^6 C _4 \times{ }^8 C _8\)</span> = 15</td>
</tr>
</tbody>
</table>
<p>Total number of ways = 1625</p>
Correct Answer: D