What is the no. of permutations of a given set of objects when repetition is allowed?

Question

Suppose we have a set of 12 alphabets. And we are required to form a 7 letter word using the 12 alphabets. Now, since repetition is allowed, the total no. of arrangements would be $\ 12^7$. The total no. of arrangements without any repetition would be $\ P_7^{12}$. From these two results, we can deduce the no. of arrangements with at least one repetition to be $\ 12^7$ - $\ P_7^{12}$. But how can we find something specific like, the total no. of arrangements such that 4 and only 4 of the letters are same (7 letter word).

Answer 1

There are 12 ways to select a letter that will be repeated exactly 4 times. We are left with 11 alphabets from which we have to choose 3 distinct letters. This can be done in $\binom{11}{3}$ ways.

Once the letters are selected, fix the fours positions of the letter that repeats. This can be done in $\binom{7}{4}$ ways. The remaining 3 distinct letters can be permuted in $3! = 6$ ways to fill the remaining three spots.

Our final answer is $12 \cdot 6 \cdot \binom{11}{3} \cdot \binom{7}{4}$.