Question

Consider the experiment of distribution of balls among urns. Suppose we are given M urns numbered 1 to M among which we are to distribute n balls (n < M). Let P(A) denote the probability that each of the urns numbered 1 to n will contain exactly one ball. Then if the balls are identical and any number of balls can go to any urns, then P(A) equals

• A. $\frac{1}{M^n}$
• B. $\frac{1}{{}^{M+n-1}{C}_{M-1}}$
• C. $\frac{1}{{}^{M+n-1}{C}_{n-1}}$
• D. $\frac{1}{{}^{M+n-1}{P}_{M-1}}$

Answer 1

Answer: (B)

$\frac{1}{{}^{M+n-1}{C}_{M-1}}$

Required no. is same as the no. of ways in which $M$ can be distributed $n$ balls and this no. is equal to coefficient of $x^M$ in ${(1+x+x^2+.....+x^M)}^n$

$P\left(A\right)$ is denoted as the probability that each of the urn number from $1$ to $n$ will contain exactly one ball.

$=$ Coefficient of $x^M$ in ${\left(\frac{1-x^{M+1}}{1-x}\right)}^n$

$=$ Coefficient of $x^M$ in ${(1-x^{M+1})}^n{(1-x)}^{-n}$

$=$ Coefficient of $x^M$ in ${(1-x)}^{-n}$

$=\left(\frac{1}{{}^{M+n-1}{C}_{M-1}}\right).$

Hence, the answer is $\left(\frac{1}{{}^{M+n-1}{C}_{M-1}}\right).$