# Math Help - subset

1. ## subset

A=(1,2,3.....30) How many 3 element subsets of A are there such that sum of its elements is divisible by 3?

Like (1,3,5) or (6,9,3)

2. ## Re: subset

Hello, kastamonu!

$A\:=\:$$1,2,3, \cdots 30$$$
How many 3-element subsets of $A$ are there
such that sum of its elements is divisible by 3?

Consider the three subsets of $A.$

Set $S_1$: multiples of 3.
. . $S_1\:=\:\{3,6,9,12,15,18,21,24,27,30\}$

Set $S_2$: one more than a multiple of 3.
. . $S_2 \;=\:\{1,4,7,10,13,16,19,22,25,28\}$

Set $S_3$: one less than a multiple of 3.
. . $S_3 \:=\:\{2,5,8,11,14,17,20,23,26,29\}$

There are four ways to get 3 elements whose sum is a multiple of 3.

[1] 3 from $S_1\!:\;{10\choose3} \,=\,120$ ways. .*

[2] 3 from $S_2\!:\;{10\choose3} \,=\,120$ ways. .*

[3] 3 from $S_3\!:\;{10\choose3} \,=\,120$ ways. .*

[4] One from each subset: . $10\cdot10\cdot10 \,=\,1000$ ways. .*

Therefore: . $120 + 120 + 120 + 1000 \:=\:1360$ ways.

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*
An explanation is available upon request.

many thanks.