Answer:
a.5005
b.![\frac{1960}{5005}](https://tex.z-dn.net/?f=%5Cfrac%7B1960%7D%7B5005%7D)
c.1/715
d.714/715
Step-by-step explanation:
We are given that
Total men=8
Total women=7
Total people, n=8+7=15
r=6
a.
Combination formula:
Selection of r out of n people by total number of ways
![nC_r](https://tex.z-dn.net/?f=nC_r)
Using the formula
We have n=15
r=6
Total number of ways=![15C_6](https://tex.z-dn.net/?f=15C_6)
Total number of ways=![\frac{15!}{6!9!}](https://tex.z-dn.net/?f=%5Cfrac%7B15%21%7D%7B6%219%21%7D)
Using the formula
![nC_r=\frac{n!}{r!(n-r)!}](https://tex.z-dn.net/?f=nC_r%3D%5Cfrac%7Bn%21%7D%7Br%21%28n-r%29%21%7D)
Total number of ways=![\frac{15\times 14\times 13\times 12\times 11\times 10\times 9!}{6\times 5\times 4\times 3\times 2\times 1\times 9!}](https://tex.z-dn.net/?f=%5Cfrac%7B15%5Ctimes%2014%5Ctimes%2013%5Ctimes%2012%5Ctimes%2011%5Ctimes%2010%5Ctimes%209%21%7D%7B6%5Ctimes%205%5Ctimes%204%5Ctimes%203%5Ctimes%202%5Ctimes%201%5Ctimes%209%21%7D)
Total number of ways=5005
b. The probability of having exactly 3 men in the group
=![\frac{8C_3\times 7C_3}{15C_6}](https://tex.z-dn.net/?f=%5Cfrac%7B8C_3%5Ctimes%207C_3%7D%7B15C_6%7D)
Using the formula
Probability,![P(E)=\frac{favorable\;cases}{Total\;number\;of\;cases}](https://tex.z-dn.net/?f=P%28E%29%3D%5Cfrac%7Bfavorable%5C%3Bcases%7D%7BTotal%5C%3Bnumber%5C%3Bof%5C%3Bcases%7D)
The probability of having exactly 3 men in the group=![\frac{\frac{8!}{3!5!}\times \frac{7!}{3!4!}}{5005}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cfrac%7B8%21%7D%7B3%215%21%7D%5Ctimes%20%5Cfrac%7B7%21%7D%7B3%214%21%7D%7D%7B5005%7D)
=![\frac{\frac{8\times 7\times 6\times 5!}{3\times 2\times 1\times 5!}\times \frac{ 7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!}}{5005}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cfrac%7B8%5Ctimes%207%5Ctimes%206%5Ctimes%205%21%7D%7B3%5Ctimes%202%5Ctimes%201%5Ctimes%205%21%7D%5Ctimes%20%5Cfrac%7B%207%5Ctimes%206%5Ctimes%205%5Ctimes%204%21%7D%7B3%5Ctimes%202%5Ctimes%201%5Ctimes%204%21%7D%7D%7B5005%7D)
=![\frac{56\times 35}{5005}](https://tex.z-dn.net/?f=%5Cfrac%7B56%5Ctimes%2035%7D%7B5005%7D)
The probability of having exactly 3 men in the group
=![\frac{1960}{5005}](https://tex.z-dn.net/?f=%5Cfrac%7B1960%7D%7B5005%7D)
c. The probability of all the selected people in the group are women
=![\frac{8C_0\times 7C_6}{5005}](https://tex.z-dn.net/?f=%5Cfrac%7B8C_0%5Ctimes%207C_6%7D%7B5005%7D)
The probability of all the selected people in the group are women
![=\frac{\frac{8!}{0!8!}\times \frac{7\times 6!}{6!1!}}{5005}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cfrac%7B8%21%7D%7B0%218%21%7D%5Ctimes%20%5Cfrac%7B7%5Ctimes%206%21%7D%7B6%211%21%7D%7D%7B5005%7D)
The probability of all the selected people in the group are women
![=\frac{7}{5005}=\frac{1}{715}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B7%7D%7B5005%7D%3D%5Cfrac%7B1%7D%7B715%7D)
d. The probability of having at least one man in the group
=1- probability of all the selected people in group are women
The probability of having at least one man in the group
![=1-\frac{1}{715}](https://tex.z-dn.net/?f=%3D1-%5Cfrac%7B1%7D%7B715%7D)
![=\frac{715-1}{715}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B715-1%7D%7B715%7D)
![=\frac{714}{715}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B714%7D%7B715%7D)
The probability of having at least one man in the group ![=\frac{714}{715}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B714%7D%7B715%7D)