Question

how many different arrangements of 5 letters can be formed if the first letter must be w or k (repeats of letters are allowed)?

there are \\(\square\\) different 5-letter combinations that can be formed.
(simplify your answer.)

Explanation:

Step1: Determine the number of choices for the first letter

Since the first letter must be \(W\) or \(K\), there are \(2\) choices for the first letter.

Step2: Determine the number of choices for the remaining letters

For each of the remaining \(4\) letters (second, third, fourth, and fifth), since repeats are allowed and there are \(26\) letters in the alphabet, there are \(26\) choices for each of these positions.

Step3: Use the multiplication principle

The total number of arrangements is the product of the number of choices at each position. By the multiplication principle \(N = 2\times26\times26\times26\times26\).
We can write this as \(N=2\times26^{4}\).
Calculate \(26^{4}=(26)^{2}\times(26)^{2}=676\times676 = 456976\).
Then \(N = 2\times456976=913952\).

Answer:

\(913952\)