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Math Help - subset

  1. #1
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    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)
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  2. #2
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    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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  3. #3
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    Re: subset

    many thanks.
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