How many different 4 letter permutations are there for the letters in the word toolroom?

Answer 1

120 four letter permutations if you don't allow more than one 'o' in the four letter

arrangement.

209 four letter permutations if you allow two, three and all four 'o'.

1.- Let set A = {t,l,r,m,}, and set B = {o,o,o,o}.

2.- From set A, the number of 4 letter permutations is 4P4 = 24.

3.- 3 letters from set A give 4P3 = 24, and one 'o' can take 4 different positions in the

word. That gives us 24x4 = 96 four letter permutations.

4.- In total, 24 + 96 = 120 different four letter permutations.

5.- If the other three 'o' are allowed to play, then you have 2 letters from set A that

give 4P2 = 12 permutations and two 'o' can take 4C2 = 6 position's, giving 12x6 = 72

four letter permutations.

6.- One letter from set A we have 4P1 = 4, each one can take 4 different positions, the

rest of the spaces taken by three 'o' gives 4x4 = 16 different permutations.

7.- The four 'o' make only one permutation.

8.- So now we get 72 + 16 + 1 = 89 more arrangements adding to a total of 89 + 120 = 209 different 4 letter arrangements made from the letters of the word toolroom.

[ nCr = n!/((n-r)!∙r!); nPr = n!/(n-r)! ]