How many $2$-digit odd numbers can be formed from the digits $1, 2, 3, 4, 5$ and $8$,if repetition of digits is allowed?

How many numbers can be formed from the digits $1, 2, 3, 4$ when the repetition is not allowed?

There are $n$ white and $n$ black balls marked $1, 2, 3, ..., n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours are

If the ratio $C(2n, 3) : C(n, 3)$ is equal to $11 : 1$,find $n$.

$A$ candidate is required to answer $6$ out of $10$ questions,which are divided into two groups each containing $5$ questions. The candidate is not permitted to attempt more than $4$ questions from each group. In how many ways can the candidate make their choice?

The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is