Question

The total number of ways in which $n^2$ number of identical balls can be put in $n$ numbered boxes $(1,2,3,...,n)$ such that $i^{th}$ box contains at least $i$ number of balls is

• A. ${}^{n^2}{C}_{n-1}$
• B. ${}^{n^2-1}{C}_{n-1}$
• C. ${}^{\frac{n^2+n-1}{2}}{C}_{n-1}$
• D. None of these

Answer 1

The correct option is C ${}^{\frac{n^2+n-1}{2}}{C}_{n-1}$

Given that there are $n^2$ identical balls that should be kept in $n$ numbered boxes,such that $ith$ box has at least $i$ balls,

Let box 1 be $a_1$,

box 2 be $a_2$, and so on,

$a_1+a_2+a_3+...+a_n=n^2$,

As $ith$ box should have at least $i$ balls we replace,

$a_1$ by $a_1+1$,

$a_2$ by $a_2+2$,

so on,

$a_n$ by $a_n+n$,

Now the equation transforms to

$(a_1+1)+{(a}_2+2)+(a_3+3)+...+{(a}_n+n)=n^2⟹a_1+a_2+a_3+...+a_n=n^2-\frac{n(n+1)}{2}=\frac{n^2-n}{2}$,

$\therefore$ the number of solutions to it are,

${}^{\frac{n^2-n-2}{{}^2{C}_{n-1}}}$