From a group of ^@8^@ men and ^@7^@ women, 5 persons are to be selected to form a committee so that at least ^@4^@ men are there on the committee. In how many ways can it be done?


Answer:

^@546^@

Step by Step Explanation:
  1. We are required to select a ^@5^@-member committe from a group of ^@8^@ men and ^@7^@ women with at least ^@4^@ men. This can be done in ^@2^@ different ways:-
    1. ^@4^@ men and ^@1^@ woman
    2. ^@5^@ men
  2. Number of ways of selecting ^@4^@ men and ^@1^@ woman ^@= 8 C _4 \times 7 C _1^@ ^@ = \dfrac{ 8! }{ 4!(8 - 4)! } \times \dfrac{ 7! }{ 1!(7 - 1)! } = 490^@
    Number of ways of selecting ^@5^@ men ^@= 8 C _5 ^@ ^@ = \dfrac{ 8! }{ 5!(8 - 5)! } = 56^@
  3. Now, required number of ways of selecting a ^@5^@-member committee ^@ = 490 + 56 = 546^@
    Hence, there are ^@546^@ ways of selecting a ^@5^@-member committee with at least ^@4^@ men.

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