The number of ways in which $5$ boys and $3$ girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never sit adjacent to each other,is

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty?

How many different four-letter words can be formed (the words need not be meaningful) using the letters of the word $MEDITERRANEAN$ such that the first letter is $E$ and the last letter is $R$?

$A$ student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select at least one book is $63$,then the value of $n$ is

There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls,one in each box,could be placed such that a ball does not go to the box of its own colour is

There are $3$ sections in a question paper and each section contains $5$ questions. $A$ candidate has to answer a total of $5$ questions,choosing at least one question from each section. Then the number of ways,in which the candidate can choose the questions,is