2020 AIME I Problems/Problem 7

Problem

A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N.$

Solution 1

We will be selecting girls, but not selecting boys. We claim that the amount of girls selected and the amount of guys not selected adds to $12$ . This is easy to see: if $k$ women were chosen, then $k + (11 - k + 1) = 12$ . Therefore, we simply take $\binom{23}{12} \implies \boxed{081}$ . ~awang11's sol

Solution 2 (Bash)

We casework on the amount of men on the committee.

If there are no men in the committee, there are $\dbinom{12}{1}$ ways to pick the women on the committee, for a total of $\dbinom{11}{0} \cdot \dbinom{12}{1}$ . Notice that $\dbinom{11}{0}$ is equal to $\dbinom{11}{11}$ , so the case where no men are picked can be grouped with the case where all men are picked. When all men are picked, all females must also be picked, for a total of $\dbinom{12}{12}$ . Therefore, these cases can be combined to $\dbinom{11}{0} \cdot (\dbinom{12}{1} + \dbinom{12}{12})$ Since $\dbinom{12}{12} = \dbinom{12}{0}$ , and $\dbinom{12}{0} + \dbinom{12}{1} = \dbinom{13}{1}$ , we can further simplify this to $\dbinom{11}{0} \cdot \dbinom{13}{1}$

All other cases proceed similarly. For example, the case with one men or ten men is equal to $\dbinom{11}{1} \cdot \dbinom{13}{2}$ . Now, if we factor out a $13$ , then all cases except the first two have a factor of $121$ , so we can factor this out too to make our computation slightly easier. The first two cases (with $13$ factored out) give $1+66=67$ , and the rest gives $121(10+75+270+504) = 103,939$ . Adding the $67$ gives $104,006$ . Now, we can test for prime factors. We know there is a factor of $2$ , and the rest is $52,003$ . We can also factor out a $7$ , for $7,429$ , and the rest is $17 \cdot 19 \cdot 23$ . Adding up all the prime factors gives $2+7+13+17+19+23 = \boxed{081}$ .

Solution 3 (Vandermonde's identity)

Applying Vandermonde's identity by setting $m=12$ , $n=11$ , and $r=11$ , we obtain $\binom{23}{11}\implies$ $\boxed{081}$ . ~Lcz

2020 AIME I (Problems • Answer Key • Resources)
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2020 AIME I Problems/Problem 7

Contents

Problem

Solution 1

Solution 2 (Bash)

Solution 3 (Vandermonde's identity)

See Also