How many words can be formed by using the letters of the word ‘DAUGHTER’ when :

(a) the vowels come together?

(b) the vowels never come together?

In how many ways can 5 children be arranged so that two of them, Ram and Rattan

(a) are always together?

(b) are never together?

5 boys and 4 girls are to be arranged in a row so that no two girls are together. In howmany can this be done?

In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if the:
(i) words start with P and end with S?
(ii) vowels are all together ?
(iii) there are always 4 letters between P and S?

How many words can be formed by using the letters of word ‘VOWELS’ if 
(a) there is no restriction on letters?
(b) each word begins with E?
(c) each word begins with E and ends with L?
(d) the vowels are together?
(e) all consonants come together?

In how many ways can a football team of 11 players be selected from 16 players if:(a) two players are always included?
(b) two players are always excluded?

(a) diagonals (b) triangles formed in a decagon.