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Chapter 1: Problem 16

An alphabet of 40 symbols is used for transmitting messages in a communication system. How many distinct messages (lists of symbols) of 25 symbols can the transmitter generate if symbols can be repeated in the message? How many if 10 of the 40 symbols can appear only as the first and/or last symbols of the message, the other 30 symbols can appear anywhere, and repetitions of all symbols are allowed?

Short Answer

Expert verified
Without restrictions, the transmitter can generate \(40^{25}\) distinct messages. With the given restrictions, the transmitter can generate \(40^2 * 30^{23}\) distinct messages.

Step by step solution

01

Calculate Number of Messages Without Restrictions

Each position in the message (of which there are 25) can be any one of the 40 symbols. Since repetitions are allowed, we can conceptualize the problem as 25 independent choices, each with 40 options. Therefore, the number of possible messages is computed as \(40^{25}\).
02

Compute Number of Messages With Restrictions

In this case, the first and the last symbols can be any of the 40 symbols (since repetitions are allowed), while the remaining 23 positions can only be any of the remaining 30 symbols. Thus the total number of distinct messages under this restriction is computed as \(40^2 * 30^{23}\).

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