Question

How many distinct 9-letter words starting with “i” can be formed from the word “committee”?

Answer 1

We have already chosen "i" to be the first letter. This means that we still have 8 letters that we can use as the second letter. Once that letter is chosen, we still have 7 letters that we can use as the third letter, and so on. This means we have

$8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 8!$

ways of choosing the remaining 8 letters.

However, some of the 8 available letters are equal: we have 2 m's, 2 t's and 2 e's. This means that we overcounted the number of words on the reasoning above. We can correct the counting by dividing three times by 2!, one time for each of the repeated letters.

The final answer is

$\dfrac{8!}{2! \times 2! \times 2!} = \dfrac{40320}{8} = 5040.$