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# Counting

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From a group of 7 men and 6 women 5 persons are to be selected to form a committee so that at least 3 men are in a committee. In how many ways can this be done.​

Aug 19, 2021

#1
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Case 1: 3 men: 7 C 3 * 6 C 2 = 525

Case 2: 4 men: 7 C 4 * 6 C 1 = 210

Case 3: 5 men: 7 C 5 = 21

525 + 210 + 21 = 756.

Aug 19, 2021
#2
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\begin{align*} \binom{7}{3}\binom{6}{2} + \binom{7}{4}\binom{6}{1} + \binom{7}{5} &= 35 \cdot 15 + 35 \cdot 6 + 21 \\ &= 7 \cdot (75+30+3) \\ &= 7 \cdot 108 \\ &= \boxed{756} \end{align*}

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Aug 19, 2021