A committee of 5 members is to be formed by selecting out of 4 men and 5 women. In how many different ways the committee can be formed if it should have at least 1 man?
A. 115
B. 120
C. 125
D. 140
E. None of these
Answer: Option C
Solution(By Examveda Team)
The committee should have(1 man, 4 women) or (2 men, 3 women) or (3 men, 2 women) or ( 4 men, 1 woman)
Required number of ways
$$ = \left( {{}^4{C_1} \times {}^5{C_4}} \right) + \left( {{}^4{C_2} \times {}^5{C_3}} \right)$$ $$ + \left( {{}^4{C_3} \times {}^5{C_2}} \right)$$ $$ + \left( {{}^4{C_6} \times {}^5{C_1}} \right)$$
$$ = \left( {{}^4{C_1} \times {}^5{C_1}} \right) + \left( {{}^4{C_2} \times {}^5{C_2}} \right)$$ $$ + \left( {{}^4{C_1} \times {}^5{C_2}} \right)$$ $$ + \left( {{}^4{C_4} \times {}^5{C_1}} \right)$$
$$ = \left( {4 \times 5} \right) + \left( {\frac{{4 \times 3}}{{2 \times 1}} \times \frac{{5 \times 4}}{{2 \times 1}}} \right)$$ $$ + \left( {4 \times \frac{{5 \times 4}}{{2 \times 1}}} \right)$$ $$ + \left( {1 \times 5} \right)$$
$$ = \left( {20 + 60 + 40 + 5} \right)$$
$$ = 125$$
Related Questions on Permutation and Combination
A. 3! 4! 8! 4!
B. 3! 8!
C. 4! 4!
D. 8! 4! 4!
A. 7560,60,1680
B. 7890,120,650
C. 7650,200,4444
D. None of these
A. 8 × 9!
B. 8 × 8!
C. 7 × 9!
D. 9 × 8!
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