Question

Assertion :$\left(n^2\right)!/(n!{)}^n$ is a natural number of all $nϵN$. Reason: Number of ways in which $n^2$ objects can be distributed among n persons equally is $\left(n^2\right)!/(n!{)}^n$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Number of ways of dividing $n^2$ objects into n groups of same size is $\frac{\left(n^2\right)!}{(n!{)}^{n}n!}$
Now number of ways of distributing these n groups among n persons is
$\left[\frac{\left(n^2\right)!}{(n!{)}^{n}n!}\right]n!=\frac{\left(n^2\right)!}{(n!{)}^n}$
which is always an integer.
Also, we know that product of r is divisible by r! Now,
$\left(n^2\right)!=1\times 2\times 3\times \mathrm{4...}{n}^2$
$=1\times 2\times 3....n$
$\times (n+1)(n+2)....2n$
$\times (2n+1)(2n+2)...3n$
$\times (n^2-(n^2-1\left)\right)(n^2-(n^2-1)...n^2$
Thus, in $n^2!$ there are n rows each consisting product of n integers. Each row is divisible by n!
Hence, both statements are correct and statement 2 is correct explanation of statement 1.