How to find the number of four letter words that can be formed with letter a,b,c such that all three letters occur

Do we need to make cases :

Case I : a,b,c,a

Case II : a,b,c,a

Case III : a,b,c,c

Please guide what is the correct approach to solve this.

## Best Answer

They way I'd go about this is as follows. First, we choose one of the three letters to be the letter that is repeated. That's three possibilities.

Next, we find all distinct permutations of our four letters. There is a letter that appears twice, and two other letters that each appear once. To count these permutations, we use the multinomial coefficient $\binom{4}{2,1,1} = \frac{4!}{2!1!1!} = 12$.

That would mean the total is $3\times 12 = 36$.