How many different four letter combinations can be made using only four letters?

24
Call the letters 1, 2, 3 and 4 you get the following combinations:
1)1234
2)1243
3)1324
4)1342
5)1423
6)1432
7)2134
8)2143
9)2314
10)2341
11)2413
12)2431
13)3124
14)3142
15)3214
16)3241
17)3412
18)3421
19)4123
20)4213
21)4231
22)4312
24)4321
Each letter as the starter gets six variations of the other three letters to go with it.

Do you mean using exactly four letters or at most four letters (four letter sequences with repetition)?

In the first situation, it's P(26, 4) = 358,800 (this is the notation for the number of 4-permutations on an alphabet of 26 letters). We can get this in two ways:

1. We have 26 choices for the first letter. Because our four letter combination is a sequence without repetition, we have 25 choices for the second letter. Then 24 for the third and 23 for the fourth. Hence we have (26)(25)(24)(23) = 358,800 such sequences.

2. First choose the 4 letters we want to use. This can be done in 26 choose 4, or 14,950, ways.

(The formula for combinations can be found at https://en.wikipedia.org/wiki/Combination )

Now order the chosen letters. There are 4! = 24 ways to do this.

(More on the factorial at https://en.wikipedia.org/wiki/Factorial )

There are thus (14,950)(24) = 358,800 such sequences.

If it's the second situation (with repetition), then it's just 26^4 = 456,976.

More on permutations at https://en.wikipedia.org/wiki/Permutation

Of course, if you already have the four letters in mind, then mehaul's answer is absolutely right. (Unless you are including sequences with repetition among the four letters, in which case the answer is 4^4 = 256. Sorry for making this more complicated than necessary!)