Question

Acomittee of c people is chosen from a group consisting of m males and f females. what is the probability that mary and peter will not be serving together?

Answer 1

the probability that Mary and Peter will not be serving together:

probability that Mary is in a committee and Peter is not in a committee +

probability that Peter is in a committee and Mary is not in a committee +

probability that Peter and Mary both are not in a committee

probability that Mary is in a committee and Peter is not in a committee:

$P_1=1/C^{c-1}_{m+f-1}=\frac{(c-1)!(m+f-c)!}{(m+f-1)!}$

probability that Mary is in a committee and Peter is not in a committee:

$P_2=P_1=1/C^{c-1}_{m+f-2}=\frac{(c-1)!(m+f-c)!}{(m+f-1)!}$

probability that Peter and Mary both are not in a committee:

$P_3=1/C^{c}_{m+f-2}=\frac{c!(m+f-c-2)!}{(m+f-2)!}$

Answer:

$P=P_1+P_2+P_3=\frac{2(c-1)!(m+f-c)!}{(m+f-1)!}+\frac{c!(m+f-c-2)!}{(m+f-2)!}$