Q8. (a). How many different 5 persons committees can be formed each containing at least two women and at least one man from a set of 10 women and 15 men. (3 Marks)

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(b). How many ways are there to distribute or district object into 12 distinct boxes with

i) At least three empty box.

ii) No Empty box.                          (4 Marks)

Solution:

Let U be all possible distributions of or distinct object into 12 boxes into 12 boxes. Let A_i denote the set of possible distributions with the set of possible distributions with the i^th box being empty.

Then the required number of distribution with at least one empty box is |A₁ U A₂ U ………..U A₁₂| We have N = 12^r. Also, |Ai| = (12 - 1)^r , the number of distribution in witch the objects are put into of the remaining four boxes.

Similarly, |A_i ∩ A_j| = (12 - 2)^r , and so eleventh. Thus by the corollary above , we have

|A₁ U A₂ U ……… U A₁₂| = S₁ – S₂ + S₃ – S₄ + S₅ – S₆ + S₇ – S₈ + S₉ – S₁₀ + S₁₁ – S₁₂

= C (12,1)11^r – C(12,2)10^r + C(12,3)9^r - C (12,4)8^r + C(12,5)7^r - C(12,6)6^r + C (12,7)5^r – C(12,8)4^r + C(12,9)3^r - C (12,10)2^r + C(12,11)1^r – 0

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