If the best and the worst paper never appear together,then six examination papers can be arranged in how many ways?

The total number of words that can be formed using the letters $a, b, c, d, e, f$ taken $3$ at a time such that each word contains at least one vowel is:

In a shop,there are $5$ types of ice-creams available. $A$ child buys $6$ ice-creams.
Statement-$1$: The number of different ways the child can buy the $6$ ice-creams is $^{10}C_5$.
Statement-$2$: The number of different ways the child can buy the $6$ ice-creams is equal to the number of different ways of arranging $6$ $A$'s and $4$ $B$'s in a row.

How many $3$-letter words can be formed from the letters of the word '$EXPERTISE$',if repetition of letters is not allowed?

In how many different ways can the letters of the word '$OPTICAL$' be arranged so that the vowels always come together?

In how many ways can the letters of the word '$UNIVERSAL$' be arranged? In how many of these will $E, R, S$ always occur together?