Question

M telegrams are distributed at random over N communication channels (N>M). Find the probability that not more than telegram will be sent over each channel.

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

Answer 1

Answer: (B)

$\frac{{}^N{C}_{M}M!}{N^M}$

Number of channels $=N$
Number of telegrams $=M$
Total number of ways of distributing $M$ telegrams over $N$ channels$=N^M$
The number of ways of choosing $M$ channels out of $N$ in order to send one telegram over each channel is ${}^N{C}_M$ and corresponding to each of these ways of choosing $M$ channels, there are $M!$ ways of sending the M telegrams.
Hence , the favorable number of ways$={}^N{C}_{m}M!$
$\therefore$ The required probability $=\frac{{}^N{C}_{M}M!}{N^M}.$