How many different words can be formed by using all the letters of the word ALLAHABAD? In how many of them vowels occupy the even positions? In how many of them both L do not come together?

Explanation

(i) The given word, 'ALLAHABAD' contains 9 letters consisting of 4 A's, 2 L's, 1 H, 1 B and 1 D.
Hence, the number of difference words formed by using all the leters of the given word `=(9!)/((4!)xx(2!))=7560.`
(ii) Let us take both L together and we treat LL as 1 letter.
Then, we will have to arrange 8 letters, namely LL, 4 A's, 1 H, 1 B and 1 D.
So, the number of words having both L together `=(8!)/(4!)=1680.`
Hence, the number of words with both L not occurring together
`=(7560-1680)=5880.`
(iii) There are 4 vowels and all are alike, i.e., 4 A's.
Also, there are 4 even places, namely 2nd, 4th, 6th and 8th.
So, we arrange A' s at these 4 places, as shown below.
`square A square A square A square A square`
Number of arrangements of 4 A's at 4 places `=(4!)/(4!)=1.`
Now, we are left with 5 letters consisting of 2 L's, 1 H, 1 B and 1 D.
Number of arrangements of these 5 letters at 5 places `=(5!)/(2!)=60.`
Hence, the number of words in which vowels occupy even places `=(1xx60)=60.`