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. Chance that not more than one telegram will be sent over each channel is:

Options

A. `(.^(N)C_(M)xxN!)/(M^(N))`
B. `(.^(N)C_(M)xxM!)/(N^(M))`
C. `1-(.^(N)C_(M)xxM!)/(M^(N))`
D. `1-(.^(N)C_(M)xxN!)/(N^(M))`

Explanation

To solve the problem of finding the probability that not more than one telegram will be sent over each channel when M telegrams are sent over N channels (where N > M), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have M telegrams and N channels.
- Each telegram can be sent over any of the N channels with equal probability.
- We need to find the probability that no channel receives more than one telegram.

2. **Total Ways to Send M Telegrams**:
- Each of the M telegrams can be sent over any of the N channels.
- Therefore, the total number of ways to send M telegrams is given by:
\[ N^M \]
- This is because for each telegram, we have N choices.

3. **Favorable Outcomes**:
- We want to find the number of ways to send M telegrams such that no channel receives more than one telegram.
- Since N > M, we can choose M channels from the N available channels. The number of ways to choose M channels from N is given by:
\[ \binom{N}{M} \]
- After choosing M channels, we can arrange the M telegrams in those M channels. The number of arrangements of M telegrams is given by:
\[ M! \]
- Therefore, the total number of favorable outcomes (ways to send M telegrams such that no channel receives more than one telegram) is:
\[ \binom{N}{M} \times M! \]

4. **Calculating the Probability**:
- The probability \( P \) that not more than one telegram will be sent over each channel is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{\binom{N}{M} \times M!}{N^M} \]

5. **Final Expression**:
- Thus, the final expression for the probability is:
\[ P = \frac{\binom{N}{M} \times M!}{N^M} \]