Question

There is a group of $20$ people that has $12$ girls and $8$ boys. A team of $6$ players is to be formed from this group, having $3$ girls and $3$ boys. In how many ways can such a group be formed?

• A. $11230$
• B. $12130$
• C. $1230$
• D. $12320$

Answer 1

The correct option is D $12320$

Given that: Number of boys $=8$

Number of girls $=12$

We need to select $3$ boys out of $8$ and $3$ girls out of $12$.

So, total number of ways
$={}^8{C}_3\times {}^{12}{C}_3$
$=\frac{8!}{(8-3)!\times 3!}\times \frac{12!}{(12-3)!\times 3!}$
$=\frac{8!}{5!\times 3!}\times \frac{12!}{9!\times 3!}$
$=\frac{8\times 7\times 6\times 5!}{5!\times 3\times 2}\times \frac{12\times 11\times 10\times 9!}{9!\times 3\times 2}$

$=(8\times 7)\times (4\times 11\times 5)$

$=56\times 20\times 11$

$=1120\times 11$

$=12320$

Short cut Method to get ${}^8{C}_3\times {}^{12}{C}_3$
${}^8{C}_3\times {}^{12}{C}_3$
$=\frac{8\times 7\times 6}{3\times 2}\times \frac{12\times 11\times 10}{3\times 2}$

$=8\times 7\times 11\times 20$

$=56\times 220$

$=12320$

Answer:

Total number of ways is $12320$.