JEE MAIN - Mathematics (2019 - 9th April Morning Slot - No. 21)
A committee of 11 members is to be formed from
8 males and 5 females. If m is the number of ways
the committee is formed with at least 6 males and
n is the number of ways the committee is formed
with at least 3 females, then :
n = m – 8
m = n = 78
m + n = 68
m = n = 68
Explanation
At least 6 males means in the committee there can be 6 males or 7 males or 8 males.
$$ \therefore $$ m = $${}^8{C_6} \times {}^5{C_5} + {}^8{C_7} \times {}^5{C_4} + {}^8{C_8} \times {}^5{C_3}$$ = 78
At least 3 females means in the committee there can be 3 females or 4 females or 5 females.
$$ \therefore $$ n = $${}^5{C_3} \times {}^8{C_8} + {}^5{C_4} \times {}^8{C_7} + {}^5{C_5} \times {}^8{C_6}$$ = 78
So, m = n = 78
$$ \therefore $$ m = $${}^8{C_6} \times {}^5{C_5} + {}^8{C_7} \times {}^5{C_4} + {}^8{C_8} \times {}^5{C_3}$$ = 78
At least 3 females means in the committee there can be 3 females or 4 females or 5 females.
$$ \therefore $$ n = $${}^5{C_3} \times {}^8{C_8} + {}^5{C_4} \times {}^8{C_7} + {}^5{C_5} \times {}^8{C_6}$$ = 78
So, m = n = 78
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