Question

How many 4 letter permutations can be made from the set {O, R, A, N, G, E} so that they all start with the letter O, and repetition of a letter is not allowed?

Answer 1

All words should start with the letter O. So the letter O is permanent at the first place of the word. Thus there's only 1 outcome at first.

It is ignored, and then let's have a look at the next three places (The word is of four letters).

There are 5 letters remaining in the set. As letter repetition is not occurred and O is permanent at the first place, O can't take the other three places.

Thus the other 5 letters R, A, N, G and E have only the chance to take the other three places of the word.

How many ways can these 5 letters be arranged in the other 3 places? Isn't it ⁵P₃ ?!

$\displaystyle \mathsf{\ ^5P_3 \ =\ \frac{5!}{(5-3)!}\ = \ \frac{5!}{2!}\ = \ 5 \times 4 \times 3\ = \ } \large \textbf{60}$

Thus we can make 60 words from the set of letters without letter repetition.