Questions: How many different arrangements of 7 letters can be formed if the first letter must be W or K (repeats of letters are allowed)? There are different 7-letter combinations that can be formed. (Simplify your answer.)

How many different arrangements of 7 letters can be formed if the first letter must be W or K (repeats of letters are allowed)?

There are different 7-letter combinations that can be formed.
(Simplify your answer.)

Transcript text: How many different arrangements of 7 letters can be formed if the first letter must be W or K (repeats of letters are allowed)? There are $\square$ different 7 -letter combinations that can be formed. (Simplify your answer.)

Solution

Solution Steps

Step 1: Identify the constraints

The problem states that the first letter must be either W or K. This means there are 2 choices for the first letter.

Step 2: Determine the number of choices for each subsequent letter

For the remaining 6 letters, there are no restrictions, and repeats are allowed. Since there are 26 letters in the English alphabet, each of the 6 positions has 26 possible choices.

Step 3: Calculate the total number of arrangements

The total number of arrangements is the product of the number of choices for each position. This can be calculated as: \[ \text{Total arrangements} = 2 \times 26^6 \]

Step 4: Compute the value

First, calculate $26^6$: \[ 26^6 = 26 \times 26 \times 26 \times 26 \times 26 \times 26 = 308,915,776 \] Then multiply by 2: \[ 2 \times 308,915,776 = 617,831,552 \]