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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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04 Sep 2017, 18:03
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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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05 Sep 2017, 17:27
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, 17:27

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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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