Permutations And Combinations

Question

How many different words can be formed by using the letters of the word ‘ALLAHABAD?(a) In how many of these do the vowels occupy even positions.(b) In how many of these, the two L’s do not come together?

Solution

Number of letters in word 'ALLAHABAD' = (A → 4, L → 2, H → 1, B → 1, D → 1) = 9
Number of arrangements   =
(a) There are only four A's as vowels.

They can occupy even places (2, 4, 6, 8) in   ways
∴   Number of ways in which vowels occupying even places = 1
We are left with 5 places and letters (L → 2, H → 1, B → 1, D → 1).
Number of permutations =
Hence, total number of arrangements in which A's occupy even places
= 1 x 60 = 60.
(b) We first find the number of arrangements in which two L's are not together:
Number of arrangements in which two L's are together

Hence, the number of arrangements in which the two L's are not together
= (Total arrangements) - (the number of arrangements in which the two L's are together)
= 7560 - 1680 = 5880.