Arrangement of 5 letter words

Question

There are 26 letters in the alphabet. How many 5-letter words can you make if you can repeat letters, but cannot have two letters in a row that are the same?

My strategy: Since there are 26 letters, the words can be made by $26 . 25. 24. 25. 26$. Is this true? I have a feeling a erred somewhere.

I thought that I couldn't choose the first letter again and that I had to keep narrowing down my choice of letters. — Aspiring Mathlete, Sep 25 '14 at 12:54

score 1 · Accepted Answer · answered Sep 25 '14 at 12:50

1

Let's break it down.

The first letter obviously has 26 choices. The second letter can be any of the letters, minus the previous one, so 25 choices. The third letter can be anything except the second letter, so 25 choices, and likewise for every other letter.

Thus the number of n-length words of this type is $26\times25^{n-1}$, or in this case:

$26\times25^{4} = 10156250$.

answered Sep 25 '14 at 12:50

extremeaxe5

1,110

Thank you! Your explanation really helped me to grasp the solution. – Aspiring Mathlete Sep 25 '14 at 12:53

score 0 · Answer 2 · answered Sep 25 '14 at 12:51

I hope I understand the question. Lets choose the letters one at a time:
You have no restrictions on the first letter, so there are 26 options, but on any letter afterwards you must not choose the letter before, leaving you with 25 options.
so the total is $26*25^4$.

Arrangement of 5 letter words

2 Answers2