Question

In Exercises 5–36, express all probabilities as fractions.

Morse Codes The International Morse code is a way of transmitting coded text by using sequences of on/off tones. Each character is 1 or 2 or 3 or 4 or 5 segments long, and each segment is either a dot or a dash. For example, the letter G is transmitted as two dashes followed by a dot, as in — — •. How many different characters are possible with this scheme? Are there enough characters for the alphabet and numbers?

Short answer

The number of different characters possible is 62.

There are enough characters available to express all the letters of the alphabet and the numbers.

Step-by-step solution

Given information

Each character is expressed in terms of 1, 2, 3, 4, or 5 segments.

Each segment can be a dot or a dash.

Describe the counting rule

The number of possibilities for each place of the arrangement is counted under the provided conditions to obtain the total counts of arrangements possible.

The total number of ways possible for the occurrence of the event is the product of the individual possibilities at each step.

Compute the number

Five different segment lengths are feasible in the Morse code, leading to five cases.

Case 1: The number of segments is one

The number of characters available for each part of the segment is two (dot or dash).

The number of ways of forming a character is two.

Case 2: The number of segments is two

The number of characters available for each part of the segment is two.

The number of ways of forming a character is $2\times 2=4$.

Case 3: The number of segments is three

The number of characters available for each part of the segment is two.

The number of ways of forming a character is $2\times 2\times 2=8$.

Case 4: The number of segments is four

The number of characters available for each part of the segment is two.

The number of ways of forming a character is$2\times 2\times 2\times 2=16$

Case 5: The number of segments is five

The number of characters available for each part of the segment is two.

The number of ways of forming a character is $2\times 2\times 2\times 2\times 2=32$.

The total number of different characters possible is equal to the sum of all feasible counts in the five cases mentioned above.

$2+4+8+16+32=62$

Therefore, the total number of different characters possible is 62.

Determine the adequacy of the Morse code

The total number of alphabets is 26.

The total number of digits is 10.

The total number of characters required is $26+10=36$.

The total number of characters available using the Morse code is 62.

Thus, there are enough characters available to express all the letters of the alphabet and the numbers.