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What is the number of 7-element subsets of the set {1, 2, 3, 4, 5, 6,

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Re: What is the number of 7-element subsets of the set {1, 2, 3, 4, 5, 6, [#permalink]

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Re: What is the number of 7-element subsets of the set {1, 2, 3, 4, 5, 6, [#permalink]

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New post 05 Sep 2017, 18:27
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thailandvc wrote:
What is the number of 7-element subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} for which the sum of those 7 elements is a multiple of 3 ?

(A) 10
(B) 11
(C) 12
(D) 13
(E) 14



\(1, 4, 7\) : They have a remainder \(1\) when they are divided by \(3\).
\(2, 5, 8\) : They have a remainder \(2\) when they are divided by \(3\).
\(3, 6, 9\) : They are multiples of \(3\).

\(1 + 2 + 3 + ... + 9 = 45\). It is a multiple of \(3\).
We need to choose two numbers whose sum is a multiple of \(3\) and subtract their sum from \(45\).
There are two cases. One is choosing two numbers from \(\{ 3, 6, 9 \}\) and other case is choosing one number from \(\{ 1, 4, 7 \}\) and one number from \(\{ 2, 5, 8 \}\).
The number of ways to choose two numbers from \(\{ 3, 6, 9 \}\) is \({}_3C_2 = 3\). And the number of ways to choose one number from \(\{ 1, 4, 7 \}\) and one number from \(\{ 2, 5, 8 \}\) is \({}_3C_1 * {}_3C_1 = 3*3 = 9\).
The number of all cases is \(3 + 9 = 12\).

The answer is C.
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Re: What is the number of 7-element subsets of the set {1, 2, 3, 4, 5, 6,   [#permalink] 05 Sep 2017, 18:27
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