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)
Hello, kastamonu!
$\displaystyle A\:=\:\(1,2,3, \cdots 30\)$
How many 3-element subsets of $\displaystyle A$ are there
such that sum of its elements is divisible by 3?
Consider the three subsets of $\displaystyle A.$
Set $\displaystyle S_1$: multiples of 3.
. . $\displaystyle S_1\:=\:\{3,6,9,12,15,18,21,24,27,30\}$
Set $\displaystyle S_2$: one more than a multiple of 3.
. . $\displaystyle S_2 \;=\:\{1,4,7,10,13,16,19,22,25,28\}$
Set $\displaystyle S_3$: one less than a multiple of 3.
. . $\displaystyle S_3 \:=\:\{2,5,8,11,14,17,20,23,26,29\}$
There are four ways to get 3 elements whose sum is a multiple of 3.
[1] 3 from $\displaystyle S_1\!:\;{10\choose3} \,=\,120$ ways. .*
[2] 3 from $\displaystyle S_2\!:\;{10\choose3} \,=\,120$ ways. .*
[3] 3 from $\displaystyle S_3\!:\;{10\choose3} \,=\,120$ ways. .*
[4] One from each subset: .$\displaystyle 10\cdot10\cdot10 \,=\,1000$ ways. .*
Therefore: .$\displaystyle 120 + 120 + 120 + 1000 \:=\:1360$ ways.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
*
An explanation is available upon request.