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Let $S=\{1, 2, 3, ⋯, 1000\}$ and $A$ be a subset of $S$. If the number of elements in $A$ is $201$ and their sum is a multiple of $5$, then $A$ is called good. How many good As are there?

My idea is bijection. The modular of subset has equal probability of $0,1,2,3,4$. So perhaps the answer is $1/5$ of choosing $201$ from $1000$?

nonuser
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Curious George
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2 Answers2

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Yes, the answer is $\frac{1}{5}\binom{1000}{201}$. Use the fact that $\gcd(201,5)=1$ to show that if $0\leq r\leq 4$ and $A_r$ is the set of subsets in $S$ with $201$ elements such that the sum of its elements is congruent to $r$ modulo $5$ then there is a bijection between them and therefore $$|A_0|=|A_1|=|A_2|=|A_3|=|A_4|.$$ Hint to construct the bijection: if $\{a_1,\dots,a_{201}\}\in A_r$ consider the set $$\{f(a_1+1),\dots,f(a_{201}+1)\}\in A_{(r+201)_5}$$ where $f(k)$ is the smallest positive integer congruent to $k$ modulo $1000$.

Robert Z
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Hint:

Use the generating function and the root of unity filter.

Look here: the root of unity filter

nonuser
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    I found this http://zacharyabel.com/papers/Multi-GF_A06_MathRefl.pdf particularly helpful from one of the answers your link provided. I'm not exactly familiar with complex numbers (8th grader here) to fully get it but will share my answer once I feel comfortable – Curious George Mar 01 '20 at 17:59