Solutions:
(i) Words start with P and end with *:*
When we fix P at the left end and at the right end, then we are left with 10 letters.
Hence, required no. of ways = (10!)/2! = 1814400**
(ii) Vowels are all together:
There are 5 vowels in the given word, 1 E, 1 U, 1 A, 1 I, and 1 O. Since these vowels are occurring together, so consider them as one letter, and when this letter is combined with the remaining 7 letters, then we have 8 letters in all, which can be arranged in (8!)/(2!) ways. Corresponding to the arrangements, the 5 vowels can be arranged in 5! ways
Hence, required no. of ways = (8!)/(2!) 5! = 20160 5! = 2419200
(iii) There are 12 words in letter PERMUTATIONS. Out of which T is repeated twice.
Now first we need to see how many ways we can make word with 4 letter between P and .
Except P and there are total of 10 letters, so number of way of selecting them = 10C4 = 210
Also note that question is asking to place exactly 4 words between P and , but does not tells you if P has to be the first letter of has to be the first letter. So In all the above combinations, we can rotate the position of P and .
So total way = 2102 = 420
The selected 4 letters can be rotated between P and in = 4! ways
So total ways = 420 4!
Consider this 6 letter chunk (P, , and 4 letter between them) as 1 letter.
Remaining letters are 6. So in total we have 7 letters, which can be arranged in 7! ways.
So total number of ways = 7! 420 4!
Now since letter T was repeated twice, we should divide the above result by 2!.
So Total number of ways = 7! 420 * 4! / 2! = 25401600
another approach for (ii)
*There are always 4 letters between P and :
Here P and are fixed, so 10 letters are left, out of which 2 are T'
So, 10 letters out of which 2 are T' can be arranged in 10!/2! = 1814400 ways
Now letters P and can be arranged so that there are 4 letters between them, which can be done in 27 = 14 ways
Required no. of ways = 1814400 14 = 25401600**
