Permutation-Combination
Aptitude

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Q.

A committee is to be formed comprising $7$ members such that there is a simple majority of men and at least $1$ woman.

The shortlist consists of $9$ men and $6$ women. In how many ways can this committee be formed?

 A.

$3,724$

 B.

$3,630$

 C.

$4,914$

 D.

$3,824$

 Hide Ans

Solution:
Option(C) is correct

Three possibilities:

$1W+6M, 2W+5M, 3W+4M$

$⇒ ({^6C_1} × {^9C_6}) + ({^6C_2} × {^9C_5}) + ({^6C_3} × {^9C_4})$

$= \textbf{4914}$

Edit: To know more about the three possibilities of selecting a committee, check comment by Ritwik.


(2) Comment(s)


Darshan
 ()

only 3 possibilities derived,how??


Ritwik
 ()

It's very simple. as the question mentions, the committee is to be formed comprising 7 members such that there is a simple majority of men and at least 1 woman.

Now keeping men in the majority and at least 1 woman in the committee, we are left with 3 choices only.

A. 1 woman and 6 men (1W+6M)

B. 2 women and 5 men (2W+5M)

C. 3 women and 4 men (3W+4M)

Any other combination does not fall in line with the conditions mentioned in the question.