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Step-by-Step Solution
Step 1: Identify the total pool of people
There are 8 males and 5 females. We want to form a committee of 11 members.
Step 2: Define the quantities m and n
• m = Number of ways to form the committee with at least 6 males.
• n = Number of ways to form the committee with at least 3 females.
Step 3: Calculate m (at least 6 males)
The committee can have exactly 6, 7, or 8 males. The respective number of females will then be 5, 4, or 3 to make a total of 11 members.
When there are 6 males: $ \binom{8}{6} \times \binom{5}{5} $
When there are 7 males: $ \binom{8}{7} \times \binom{5}{4} $
When there are 8 males: $ \binom{8}{8} \times \binom{5}{3} $
Hence,
$$
m = \binom{8}{6} \cdot \binom{5}{5}
+ \binom{8}{7} \cdot \binom{5}{4}
+ \binom{8}{8} \cdot \binom{5}{3}.
$$
Substituting and summing up gives
$$ m = 78. $$
Step 4: Calculate n (at least 3 females)
A committee with at least 3 females can have exactly 3, 4, or 5 females. Correspondingly, the rest will be males.
When there are 3 females: $ \binom{5}{3} \times \binom{8}{8} $
When there are 4 females: $ \binom{5}{4} \times \binom{8}{7} $
When there are 5 females: $ \binom{5}{5} \times \binom{8}{6} $
Hence,
$$
n = \binom{5}{3} \cdot \binom{8}{8}
+ \binom{5}{4} \cdot \binom{8}{7}
+ \binom{5}{5} \cdot \binom{8}{6}.
$$
After evaluating, we also get
$$ n = 78. $$
Step 5: Conclusion
Therefore, both m and n evaluate to 78. The correct answer is
m = n = 78.
