statistics permutations and combinations

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Selecting a committee: There are 7 women and 5 men in a department. How many ways can a committee of 4 be selected if there must be at least 2 women?

  • statistics permutations and combinations -

    3 cases :
    2 women, 2 men ---> C(7,2)xC(5,2) = 21(10) = 210
    3 women, 1 man ---> C(7,3)xC(5,1) = 35(5) = 175
    4 women, 0 men ---> C(7,4) = 35

    add up the 4 cases to get 420

    or, exclude the case of no women, and one woman
    C(12,4) - C(7,1)xC(5,3) - C(7,0)x(C5,4)
    = 495 - 70 - 5 = 420

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