Question

In how many ways $8$ people can be divided into two groups of $5$ and $3$ people?

• A. $3136$
• B. $56$
• C. $560$
• D. $1$

Answer 1

The correct option is B $56$

Total number of people $=8$
Number of people in first group $=5$
Number of people in second group $=3$

The first group of $5$ people can be selected from $8$ people in ${}^8{C}_5$ ways.

${}^8{C}_5=\frac{8!}{5!(8-5)!}$

${}^8{C}_5=\frac{8!}{5!3!}$

${}^8{C}_5=56$


The first group of $5$ people can be selected from $8$ people in $56$ ways.

Now, we are available with $8-5=3$ people as $5$ people are already selected in the first group.

We need to select $3$ people from the $3$ available people to form the second group.

The second group of $3$ people can be selected from $3$ people in ${}^3{C}_3$ ways.

${}^3{C}_3=\frac{3!}{3!(3-3)!}$

${}^3{C}_3=\frac{3!}{3!0!}$

${}^3{C}_3=1$ ($\because 0!=1$)

Total number of ways in which $8$ people can be divided into group of $5$ and $3$ people:
$=56\times 1=56$

Therefore, option (b.) is the correct answer.