Question

A local post office is to send $M$ telegrams which are distributed at random over $N$ communication channels $(N>M)$. Each telegram is sent over any channel with equal probability. The chance that not more than one telegram will be sent over each channel is

• A. $\frac{{}^N{C}_M.M!}{N^M}$
• B. $\frac{{}^N{C}_M.N!}{M^N}$
• C. $1-\frac{{}^N{C}_M.M!}{M^N}$
• D. $1-\frac{{}^N{C}_M.N!}{N^M}$

Answer 1

Answer: (A)

$\frac{{}^N{C}_M.M!}{N^M}$

Number of telegrams = M

Number of channels = N, where (N>M)

Total number of ways = $N^M$

Number of ways = ${}^N{C}_M$

Number of ways of sending M telegrams through N channels = $M!$

Favorable ways = ${}^N{C}_M\times M!$

Hence, the probability that not more than one telegram will be sent over each channel is = $\frac{{}^N{C}_M.M!}{N^M}$