How many words/strings of length 5 can we make using the first 10 letters of the alphabet with at least one repeated letter?

Question

How would you approach a problem like this? If I were to make words of length 5 from the first 10 letters it would be 10^5 or 10x10x10x10x10, right? But how do I account for the repetition part? repeated does not mean that they have to be next to each other. It just means that the same letter exists more than once.

Answer 1

First 10 alphabets of English language are :

A,B,C,D,E,F,G,H,I,J.

So, if you want to write the 5 letter string without any repetition, then, the ways should be : $$\binom{10}{5}*5!$$ Also, atleat 1 means : TOTAL WAYS(can repeat) - NONE REPEATING.

Thus, Total ways = $10*10*10*10*10 = 10^5$

Thus, required number of ways is: $$10^5 - \binom{10}{5}*5!$$

And that's your answer!