- #1
squelch
Gold Member
- 57
- 1
The problem statement:
How many five-letter strings of capital letters have a letter repeated twice in a row? For example, include ABCCA and AAABC and ABBCC but not ABCAD.
The attempt at a solution:
How many five-letter strings of capital letters have a letter repeated twice in a row? For example, include ABCCA and AAABC and ABBCC but not ABCAD.
The attempt at a solution:
- First, let's break down how we would perform the selection of a string that meets the question's criteria.
- We select the first letter. There are 26 choices. This selection also triggers selection of the second letter, for which there is only one choice.
- We then decide where in the five-position string to place these letters. There are four possible positions.
- We then select for the other three letters. There are 26 possibilities for each letter, but we should exclude repeats. So the number of permutations for 26 letters to appear in three positions is P(26,3)=##\frac{26!}{(26-3)!}##
- Then we combine the choices using the Product Rule.
- [choices for first letter] * [choices for second letter] * [choices for position of first and second letter] * [permutations for remaining letters]
- ##26*1*4*\frac{26!}{(26-3)!}=1622400##
- We can then conclude that there are 1 622 400 such choices.