# How many words can be formed out of the letters of the word GRANDMOTHER, such that each word starts with G and ends with R?

This is a problem from a specimen question paper: "How many words can be formed out of the letters of the word GRANDMOTHER, such that each word starts with G and ends with R?"

My problem is that the question does not make it clear whether the letters can be repeated or not, and therefore, I assumed that repetition of letters is NOT allowed and this is my working:

Vowels : {A, O, E} Consonants: {G, R, N, D, M, T, H, R}

For first and last letter, we have only two choices: G and R For remaining 9 letters in between, we have 9P9 choices. Therefore, total possible words= 9P9= 362880

I would like to know if my approach and answer are correct. Also, what should I assume in regard to repetition of letters, incase questions are ambiguous as the above one?

Thanks!

• If repetition of letters were allowed, the number of possible words would be infinite. – TonyK Feb 4 '20 at 10:20

As you say, the problem is not particularly clear. What you have is correct if you are supposed to use all the letters: once you fix the G and R at the start and end, all the $$9$$ internal letters are different, so it is just the number of ways to permute them.

However, another possible interpretation is that you have to use letters from "GRANDMOTHER", but not necessarily all of them, so e.g. "GOMER" is an acceptable word.

In general I would assume that you are allowed to use at most as many copies of a given letter as there are in the original. This is especially the case here where there are two Rs and you are presumably allowed to use both - if letters couldn't be repeated, or if any letter can be used as many times as you want, there would be no need to give a word with repeated letters in the first place. Also, in this case there is no restriction given on the length of the words, so if you were allowed to repeat letters ad lib there would be infinitely many possibilities.

• I see..Thanks! I only hope the examiner agrees with my assumption, or else I have to go for cases. – user687774 Feb 4 '20 at 8:40