Question

an executive committee consists of 15 members: 7 men and 8 women. 6 members are selected at random to attend a meeting in hawaii. the names are drawn from a hat. what is the probability that all 6 selected are men?
the probability that all selected are men is
(simplify your answer. type an integer or a simplified fraction.)

Explanation:

Step1: Calculate combinations for men

We need to choose 6 men from 7. The combination formula is \( C(n, k) = \frac{n!}{k!(n - k)!} \). So \( C(7, 6)=\frac{7!}{6!(7 - 6)!}=\frac{7!}{6!1!}=\frac{7\times6!}{6!×1}=7 \).

Step2: Calculate total combinations

We need to choose 6 members from 15. Using the combination formula, \( C(15, 6)=\frac{15!}{6!(15 - 6)!}=\frac{15!}{6!9!}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}=5005 \).

Step3: Calculate probability

The probability is the number of favorable outcomes (choosing 6 men) divided by the total number of outcomes (choosing 6 members). So the probability \( P=\frac{C(7, 6)}{C(15, 6)}=\frac{7}{5005}=\frac{1}{715} \).

Answer:

\(\frac{1}{715}\)