Question
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:
A.
$$m + n = 68$$
B.
$$m = n = 78$$
C.
$$n = m - 8$$
D.
$$m = n = 68$$
Answer :
$$m = n = 78$$
Solution :
Since, $$m$$ = number of ways the committee is formed with at least 6 males
$$ = {\,^8}{C_6}.\,{\,^5}{C_5} + {\,^8}{C_7}.\,{\,^5}{C_4} + {\,^8}{C_8}.\,{\,^5}{C_3} = 78$$
and $$n$$ = number of ways the committee is formed with at least 3 females
$$ = {\,^5}{C_3}.\,{\,^8}{C_8} + {\,^5}{C_4}.\,{\,^8}{C_7} + {\,^5}{C_5}.\,{\,^8}{C_6} = 78$$
Hence, $$m = n = 78$$