$12$ persons are to be arranged at a round table. If two particular persons among them are not to be side by side,the total number of arrangements 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.

In how many different ways can the letters of the word '$LEADING$' be arranged in such a way that the vowels always come together?

If all the words,with or without meaning,are written using the letters of the word $QUEEN$ and are arranged as in an English dictionary,then the position of the word $QUEEN$ is (in $^{th}$)

$A$ bag contains balls of two colours,$3$ black and $3$ white. What is the minimum number of balls which must be drawn from the bag,without looking,so that among these there are two of the same colour?

There are $n$ distinct points on the circumference of a circle. The number of pentagons that can be formed with these points as vertices is equal to the number of possible triangles. Then the value of $n$ is