(i) How many different words can be formed by using all the letters of the word, 'ALLAHABAD'?
In how many of them :
(ii) Both L's do not come together ?
(iii) The vowels occupy the even positions ?
(i) How many different words can be formed by using all the letters of the word, 'ALLAHABAD'?
In how many of them :
(ii) Both L's do not come together ?
(iii) The vowels occupy the even positions ?
(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 different words formed by using all the letters of the given word = 9 !(4 !)×(2 !)=7560.
(ii) Let us take both L together and we treat 
as 1 letter.
Then, we will have to arrange 8 letters, namely 
, 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 place, as shown below.
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 place = (1×60)=60.
(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 different words formed by using all the letters of the given word = 9 !(4 !)×(2 !)=7560.
(ii) Let us take both L together and we treat as 1 letter.
Then, we will have to arrange 8 letters, namely , 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 place, as shown below.
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 place = (1×60)=60.
