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Assume that the committee consists of 4 Republicans and 5 Democrats. A subcommittee consisting of 5 people is to be selected.

How many such subcommittees are possible if each subcommittee must contain at least 1 and no more than 2 Republicans?

  • math -

    If C(M,N) represents the number of combinations by choosing N objects out of M, and equals M!/((M-N)!N!)

    Case 1:
    1 republican + 4 democrats
    P(1,4)
    =C(4,1)*C(5,4)
    =4!/(1!3!) * (5!/(4!1!))
    =20

    Case 2:
    2 republicans + 3 democrats
    P(2,3)
    =C(4,2)*C(5,3)
    =4!/(2!2!) * (5!/(3!2!))
    =12*10
    =120

    Total number of different committees
    =20+120
    =140

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