What is the formula for Combination with repetition?

Question

• What is the number of ways we can choose three letters from $\{A,B,...,Z\}$ if repetition is allowed?

This is not a permutation.

Is there any formula to solve this kind of problems?

Answer 1

Yes, there is a formula. Apply it with $n=26$ and $k=3$.

Answer 2

What is the number of ways we can choose three letters from {A,B,...,Z} if repetition is allowed?

English alphabet has 26 letters $\{A,B,C,...,Z\}$ and you want a string of length 3 so say we have 3 empty spots __ __ __

How to do it practically:

The first spot could have any of the $26$ letters, the second can have any of the $26$ letters, and finally the third will have any of the $26$ letters. Therefore, you have $26*26*26$ which is equal to $26^3$ since the repetition is allowed.

If elements are not to be repeated so you have $26$ different possibilities for the first slot, $25$ for the second since $1$ was used in the first, and finally, $24$ left for the last one which would give $26*25*24$ which would be permutations without repetition and with order, which is $26P3$

if the order doesn't matter and repetition is not allowed ${26 \choose 3}$

And for the formula, it is what Gibbs said.