Question

a group of mathematics faculty at the local college consists of 6 women and 10 men. five people are to be selected to go to a conference. how many different ways can a group of five people be selected from this group of 16? in how many ways can five women be chosen from the group of 6 women? what is the probability that all women will be chosen to attend the conference? enter as decimal rounded to the nearest hundredth.

Explanation:

Step1: Calculate number of ways to select 5 from 16

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 16$ and $r=5$.
$C(16,5)=\frac{16!}{5!(16 - 5)!}=\frac{16!}{5!×11!}=\frac{16\times15\times14\times13\times12}{5\times4\times3\times2\times1}=4368$

Step2: Calculate number of ways to select 5 from 6 women

Using the combination formula with $n = 6$ and $r = 5$.
$C(6,5)=\frac{6!}{5!(6 - 5)!}=\frac{6!}{5!×1!}=6$

Step3: Calculate the probability

The probability $P$ that all - women are chosen is the number of ways to choose 5 women divided by the number of ways to choose 5 people from 16.
$P=\frac{C(6,5)}{C(16,5)}=\frac{6}{4368}\approx0.0014\approx0.00$

Answer:

4368
6
0.00