The given word HAPPYHOLI has 9 letters

These 9 letters can e arranged in 9! ways.

But here in the given word letters H & P are repeated twice each

Therefore, Number of ways these 9 letters can be arranged is 

9 ! 2 ! x 2 ! = 9 x 8 x 7 x 6 x 5 x 4 x 3 2 = 90 , 720 ways .

Given Word is PROMISE.

Number of letters in the word PROMISE = 7

Number of ways 7 letters can be arranged = 7! ways

Number of Vowels in word PROMISE = 3 (O, I, E)

Number of ways the vowels can be arranged that 3 Vowels come together = 5! x 3! ways

Now, the number of ways of arrangements so that three vowels should not come together

= 7! - (5! x 3!) ways = 5040 - 720 = 4320.

The number of ways in which the letters of the word RAILINGS can be arranged such that R & S always come together is

Count R & S as only 1 space or letter so that RS or SR can be arranged => 7! x 2!

But in the word RAILINGS, I repeated for 2 times => 7! x 2!/2! = 7! ways = 5040 ways.

Total number of flowers = (8+7+6) = 21.

Let E = event that the flower drawn is neither red nor green. 

= event taht the flower drawn is blue. 

--> n(E)= 7 

--> P(E)=   7 21= 1 3 

We may have (1 boy and 3 girls)or(2boys and 2 girls)or(3 boys and 1 girl)or(4 boys).

Required number of ways = ( C 1 6 × C 3 4) +  C 2 6 × C 2 4  + ( C 3 6 × C 1 4) + ( C 4 6)  

= (24+90+80+15) 

= 209.

Given word is TRANSFORMER.

Number of letters in the given word = 11 (3 R's)

Required, number of ways the letters of the word 'TRANSFORMER' can be arranged such that 'N' and 'S' always come together is

10! x 2!/3!

= 3628800 x 2/6

= 1209600

Here in 100P2, P says that permutations and is defined as in how many ways 2 objects can be selected from 100 and can be arranged.

That can be done as,

100 P 2 = 100!/(100 - 2)!

= 100 x 99 x 98!/98!

= 100 x 99 

= 9900.

We have to arrange 6 books.

The number of permutations of n objects is n! = n. (n ? 1) . (n ? 2) ... 2.1

Here n = 6 and therefore, number of permutations is 6.5.4.3.2.1 = 720