Permutations & Combinations
Grade None
Question:
<p>A committee of 12 is to be formed from 9 women and 8 men in which at least 5 women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively,</p>
<p style="display:inline">2016, 630</p>
<p style="display:inline">4134, 56</p>
<p style="display:inline">2352, 7329</p>
<p style="display:inline">2702, 1008</p>
Step-by-Step Solution
Key Concept: Solve by identifying all mutually exclusive combinations of men and women that satisfy the total committee size, the minimum gender constraint, and the specific majority condition.
<p><strong>Case I:</strong> Women are in majority<br />
The committee contains<br />
<span class="math-tex">\(\equiv\)</span> (7W & 5M) or (8W & 4M) or (9W & 3M)<br />
<span class="math-tex">\(\Rightarrow\)</span> Number of committees that can be formed<br />
= <sup>9</sup>C<sub>7</sub> <span class="math-tex">\(\times\)</span> <sup>8</sup>C<sub>5</sub> + <sup>9</sup>C<sub>8</sub> <span class="math-tex">\(\times\)</span> <sup>8</sup>C<sub>4</sub> + <sup>9</sup>C<sub>9</sub> <span class="math-tex">\(\times\)</span> <sup>8</sup>C<sub>3</sub><br />
= 36 <span class="math-tex">\(\times\)</span> 56 + 9 <span class="math-tex">\(\times\)</span> 70 + 1 <span class="math-tex">\(\times\)</span> 56<br />
= 2016 + 630 + 56<br />
= 2702<br />
<strong>Case II:</strong> Men are in majority<br />
The committee contains 7M & 5W<br />
<span class="math-tex">\(\Rightarrow\)</span> Number of committees that can be formed<br />
= <sup>8</sup>C<sub>7</sub> <span class="math-tex">\(\times\)</span> <sup>9</sup>C<sub>5</sub><br />
= 8 <span class="math-tex">\(\times\)</span> 126 = 1008</p>
Correct Answer: D