Radio stations in a certain country use a sequence of 4 or 5 letters as their station identification call letters. The first letter must be Z. Assume there are no restrictions on the remaining letters, and repetition is allowed. a) How many 4-letter station identifications are possible? b) How many 5-letter station identifications are possible? c) How many total station identifications are possible? d) The identification for a randomly-chosen radio station is 4 letters in length. What is the probability that all four letters are different?
Radio stations in a certain country use a sequence of 4 or 5 letters as their station identification call letters. The first letter must be Z. Assume there are no restrictions on the remaining letters, and repetition is allowed. a) How many 4-letter station identifications are possible? b) How many 5-letter station identifications are possible? c) How many total station identifications are possible? d) The identification for a randomly-chosen radio station is 4 letters in length. What is the probability that all four letters are different?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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### Radio Station Identification Problem
Radio stations in a certain country use a sequence of 4 or 5 letters as their station identification call letters. The first letter must be Z. Assume there are no restrictions on the remaining letters, and repetition is allowed.
#### Questions:
a) How many 4-letter station identifications are possible?
b) How many 5-letter station identifications are possible?
c) How many total station identifications are possible?
d) The identification for a randomly-chosen radio station is 4 letters in length. What is the probability that all four letters are different?
---
### Solutions:
a) **Set up the expression that would be used to calculate the number of possible 4-letter station identifications.**
The expression is \( Z \times 26^3 \).
*Explanation:*
- The first letter must be Z.
- The remaining three letters can be any of the 26 letters in the alphabet.
*There are*
\[ 26^3 \]
*possible 4-letter station identifications.*
**Simplified Answer:** \( 17576 \)
b) **There are**
\[ 26^4 \]
*possible 5-letter station identifications.*
**Simplified Answer:** \( 456976 \)
c) **The total number of possible station identifications is:**
\[ 26^3 + 26^4 \]
*Simplified Answer:* \( 474552 \)
d) **If the identification for a randomly-chosen radio station is 4 letters in length, then the probability that all four letters are different is:**
\[ \frac{26 \times 25 \times 24}{26^3} \]
*Rounded to three decimal places as needed.*
**Simplified Answer:** \( 0.333 \)
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Transcribed Image Text:---
### Radio Station Identification Problem
Radio stations in a certain country use a sequence of 4 or 5 letters as their station identification call letters. The first letter must be Z. Assume there are no restrictions on the remaining letters, and repetition is allowed.
#### Questions:
a) How many 4-letter station identifications are possible?
b) How many 5-letter station identifications are possible?
c) How many total station identifications are possible?
d) The identification for a randomly-chosen radio station is 4 letters in length. What is the probability that all four letters are different?
---
### Solutions:
a) **Set up the expression that would be used to calculate the number of possible 4-letter station identifications.**
The expression is \( Z \times 26^3 \).
*Explanation:*
- The first letter must be Z.
- The remaining three letters can be any of the 26 letters in the alphabet.
*There are*
\[ 26^3 \]
*possible 4-letter station identifications.*
**Simplified Answer:** \( 17576 \)
b) **There are**
\[ 26^4 \]
*possible 5-letter station identifications.*
**Simplified Answer:** \( 456976 \)
c) **The total number of possible station identifications is:**
\[ 26^3 + 26^4 \]
*Simplified Answer:* \( 474552 \)
d) **If the identification for a randomly-chosen radio station is 4 letters in length, then the probability that all four letters are different is:**
\[ \frac{26 \times 25 \times 24}{26^3} \]
*Rounded to three decimal places as needed.*
**Simplified Answer:** \( 0.333 \)
---
This page has been designed to provide an understanding of identifiers used by radio stations and simple probability involving combinations of letters.
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