Question

A whole sale company has to supervise 35 areas. 7 members of the staff are available and each is to be assigned 5 areas to supervise. The number of ways the 35 areas can be put in to 7 groups of 5 and the number of ways, the areas can be assigned to the staff are respectively

• A. $\frac{35!}{{(7!)}^5},\frac{35!}{{(7!)}^5}$
• B. $\frac{35!}{{(5!)}^{7}7!},\frac{35!}{{(5!)}^7}$
• C. $\frac{5!35!}{{(5!)}^7},\frac{35!}{{(7!)}^5}$
• D. $\frac{35!}{{(5!)}^7},\frac{35!}{{(7!)}^4}$

Answer 1

The correct option is B $\frac{35!}{{(5!)}^{7}7!},\frac{35!}{{(5!)}^7}$

Fact : If $S$ is the set of $mn$ objects then $S$ can be divided into $n$ sets of $m$ elements $=\frac{\left(mn\right)}{{(m!)}^{n}n!}$ ways

In our case $m=5,n=7$
So the number of ways the area can be put into group of $5=\frac{(7\times 5)!}{{(5!)}^{7}7!}=\frac{35!}{{(5!)}^{7}7!},$
For the other part let us consider that the area have been arranged in $5$ groups of $7$ in the same particular way.
Then the $7$ groups can be assigned to $7$ men in $7!$ way depending on which group goes to the first man which to the second and similarly the other.
$\therefore$ The required number of ways in which the area can be assigned to the members/the staff is

$\frac{7!35!}{{(5!)}^{7}7!}=\frac{35!}{{(5!)}^7}$
$\therefore$ Required Number of ways are $=\frac{35!}{{(5!)}^{7}7!},\frac{35!}{{(5!)}^7}$