Answer :
To determine in how many ways the four call letters of a radio station can be arranged if the first letter must be W or K and no letters repeat, we can follow a step-by-step approach:
1. Choose the First Letter:
The first letter should be either W or K. So, there are 2 possible choices for the first letter.
2. Remaining Letters:
Once the first letter is chosen, we have used 1 of the 26 letters in the alphabet. Hence, 25 letters remain available.
3. Arranging the Remaining Three Letters:
Since no letters can repeat, we have the following choices:
- For the second position, we can choose any of the remaining 25 letters.
- For the third position, we can choose any of the remaining 24 letters after picking the second letter.
- For the fourth position, we can choose any of the remaining 23 letters after picking the third letter.
4. Calculating the Total Arrangements:
The total number of arrangements is calculated by multiplying the number of choices for each position:
- Number of ways to choose the first letter: 2
- Number of ways to choose the second letter: 25
- Number of ways to choose the third letter: 24
- Number of ways to choose the fourth letter: 23
Therefore, the total number of arrangements is:
[tex]\[ 2 \times 25 \times 24 \times 23 = 27600 \][/tex]
Thus, the four call letters of a radio station can be arranged in 27,600 different ways if the first letter must be W or K and no letters repeat.
1. Choose the First Letter:
The first letter should be either W or K. So, there are 2 possible choices for the first letter.
2. Remaining Letters:
Once the first letter is chosen, we have used 1 of the 26 letters in the alphabet. Hence, 25 letters remain available.
3. Arranging the Remaining Three Letters:
Since no letters can repeat, we have the following choices:
- For the second position, we can choose any of the remaining 25 letters.
- For the third position, we can choose any of the remaining 24 letters after picking the second letter.
- For the fourth position, we can choose any of the remaining 23 letters after picking the third letter.
4. Calculating the Total Arrangements:
The total number of arrangements is calculated by multiplying the number of choices for each position:
- Number of ways to choose the first letter: 2
- Number of ways to choose the second letter: 25
- Number of ways to choose the third letter: 24
- Number of ways to choose the fourth letter: 23
Therefore, the total number of arrangements is:
[tex]\[ 2 \times 25 \times 24 \times 23 = 27600 \][/tex]
Thus, the four call letters of a radio station can be arranged in 27,600 different ways if the first letter must be W or K and no letters repeat.
