How many words can be formed from all the letters of the word 'INITIAL' that start and end with the letter 'I'?

Question

How many words can be formed from all the letters of the word 'INITIAL' such that all words must have started and ended with letter 'I'?

Answer 1

Just fix two $I's$ in front and last $$\{I,\_,\_,\_,\_,\_,I\}$$

Now you have five objects

$$\{N,I,T,A,L\}$$

And their permutation is $$5!=120$$

Answer 2

'INITIAL' has 7 letters. You know that 2 of them have to stay at the beginning and at the end of the word, so 2 (I) letters and 2 (start and end) position are "blocked".

You need to think to the middle positions. Middle positions are 5 and you have 5 letters. So, you want all the words can be formed by the positions of the letters: you need to use disposition of 5 objects in 5 positions: PERMUTATION.

$$ D_{5,5}= P_5=5! $$