Permutations & Combinations
Selection and Committee Formation
Grade 11
Question:
<p>A committee of 11 members is to be formed from 8 males and 5 females. If <i>m</i> is the number of ways the committee is formed with at least 6 males and <i>n</i> is the number of ways the committee is formed with at least 3 females, then</p>
<p>(a) <i>m</i> = <i>n</i> = 68</p>
<p>(b) <i>m</i> + <i>n</i> = 68</p>
<p>(c) <i>m</i> = <i>n</i> = 78</p>
<p>(d) <i>n</i> = <i>m</i> - 8</p>
Step-by-Step Solution
Key Concept: Use complementary counting with binomial coefficients to find the number of ways to form a committee satisfying the given constraints on gender composition.
<p><strong>Step 1:</strong> Find <i>m</i> = number of ways when there is at least 6 males.</p><p>$$m = \binom{8}{6}\binom{5}{5} + \binom{8}{7}\binom{5}{4} + \binom{8}{8}\binom{5}{3}$$</p><p>$$= (28 \times 1) + (8 \times 5) + (1 \times 10) = 28 + 40 + 10 = 78$$</p><p><strong>Step 2:</strong> Find <i>n</i> = number of ways when there is at least 3 females.</p><p>$$n = \binom{5}{3}\binom{8}{8} + \binom{5}{4}\binom{8}{7} + \binom{5}{5}\binom{8}{6}$$</p><p>$$= 10 \times 1 + 5 \times 8 + 1 \times 28 = 78$$</p><p><strong>Conclusion:</strong> <i>m</i> = <i>n</i> = 78, so the answer is (c).</p>
Correct Answer: C