The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

How many numbers of four digits can be formed with the digits $1, 2, 3, 4$ and $5$?

Find $\sum_{r=1}^{5} C(5, r)$

$A$ 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?

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

Let $A$ and $B$ be two sets containing $4$ and $2$ elements respectively. Then the number of subsets of the set $A \times B$,each having at least $3$ elements is: