The number of integers lying between $1000$ and $10000$ such that every number contains the digits $3$ and $7$ exactly once without repetition is:

If $8 \cdot {}^{7}P_{r} = 7 \cdot {}^{8}P_{r-1}$,what is the value of $r$?

All the letters of the word $TABLE$ are permuted and the strings of letters (may or may not have meaning) thus formed are arranged in dictionary order. Then the rank of the word $TABLE$ counted from the rank of the word $BLATE$ is

The number of $4$-digit numbers that are not divisible by $5$ is:

The value of $(2 \cdot {}^{1}P_{0} - 3 \cdot {}^{2}P_{1} + 4 \cdot {}^{3}P_{2} - \dots \text{ up to } 51^{\text{th}} \text{ term}) + (1! - 2! + 3! - \dots \text{ up to } 51^{\text{th}} \text{ term})$ is equal to

$A$ linguistic club consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this group including the selection of a leader (from among these $4$ members) for the team. If the team has to include at most one boy,the number of ways of selecting the team is