Question

the inhabitants of the island of jumble use the standard roman alphabet ($26$ letters, a through z). each word in their language is $3$ letters, and for some reason, they insist that all words contain the letter a at least once. how many $3$-letter words are possible?

Answer 1

There are 1977 possible 3-letter words in the island of Jumble's language.

Step-by-step explanation:

To find the number of 3-letter words possible in the island of Jumble's language, we need to consider that all words must contain the letter a at least once. We can break down this problem into three cases:

1. Case 1: a is chosen as the first letter. In this case, we have to choose the remaining two letters from the remaining 25 letters (excluding a). There are 25 choices for each of the remaining two letters. So, the number of possibilities in this case is 25 * 25 = 625.
2. Case 2: a is chosen as the second letter. In this case, the first and third letters can be chosen from any of the 26 letters. So, the number of possibilities in this case is 26 * 1 * 26 = 676.
3. Case 3: a is chosen as the third letter. Similar to case 2, the first and second letters can be chosen from any of the 26 letters. So, the number of possibilities in this case is 26 * 26 * 1 = 676.

To find the total number of 3-letter words possible, we add up the number of possibilities in each case: 625 + 676 + 676 = 1977. Therefore, there are 1977 possible 3-letter words in the island of Jumble's language.