In the below word how many words are there in which R and W are at the end positions?

RAINBOW

When at least 2 women are included.

The committee may consist of 3 women, 2 men : It can be done in  4C3*6C2  ways

or, 4 women, 1 man : It can be done in  4C4*6C1ways

or, 2 women, 3 men : It can be done in 4C2*6C3 ways.

Total number of ways of forming the committees

= 4C2*6C3+4C3*6C2+4C4*6C1

= 6 x 20 + 4 x 15 + 1x 6

= 120 + 60 + 6 =186

Number of ways of choosing 2 black pens from 5 black pens in 5C2 ways.

Number of ways of choosing 2 white pens from 3 white pens in 3C2 ways.

Number of ways of choosing 2 red pens from 4 red pens in 4C2ways.

By the Counting Principle, 2 black pens, 2 white pens, and 2 red pens can be chosen in 10 x 3 x 6 =180 ways.

For any set of 4 points we get a cyclic quadrilateral. Number of ways of choosing 4 points out of 12 points is 12C4 = 495.

Therefore, we can draw 495 quadrilaterals

There are 7 letters in the word Bengali of these 3 are vowels and 4 consonants.

There are 4 odd places and 3 even places. 3 vowels can occupy 4 odd places in 4P3 ways and 4 constants can be arranged in 4P4 ways.

Number of words =4P3  x 4P4= 24 x 24 = 576

Here the order of choosing the elements doesn’t matter and this is a problem in combinations.

We have to find the number of ways of choosing 4 elements of this set which has 11 elements.

This can be done in 11C4 ways = 330 ways

There are 7 letters in the word ‘Bengali of these 3 are vowels and 4 consonants.

Considering vowels a, e, i as one letter, we can arrange 4+1 letters in 5! ways in each of which vowels are together. These 3 vowels can be arranged among themselves in 3! ways.

Total number of words = 5! x 3!= 120 x 6 = 720

Let 4 girls be one unit and now there are 6 units in all.

They can be arranged in 6! ways.

In each of these arrangements 4 girls can be arranged in 4! ways. 

Total number of arrangements in which girls are always together = 6! x 4!= 720 x 24 = 17280

There are 4 books on fairy tales and they have to be put together. They can be arranged in 4! ways.

Similarly, there are 5 novels.They can be arranged in 5! ways.

And there are 3 plays.They can be arranged in 3! ways.

So, by the counting principle all of them together can be arranged in 4!´5!´3! ways = 17280