If $4$-digit numbers greater than $5,000$ are randomly formed from the digits $0, 1, 3, 5,$ and $7$,what is the probability of forming a number divisible by $5$ when the repetition of digits is not allowed?

The domain of the function $f(x) = ^{16 - x}C_{2x - 1} + ^{20 - 3x}P_{4x - 5}$,where the symbols have their usual meanings,is the set

If $_nP_r = 30240$ and $\binom{n}{r} = 252$,then $(n, r) = \dots$

The total number of three-digit numbers,where exactly one digit is repeated two times,is

For integers $n$ and $r$,let $\binom{n}{r} = \begin{cases} ^{n}C_{r}, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$. The maximum value of $k$ for which the sum $\sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i}$ exists,is equal to ...... .

From a group of $7$ batsmen and $6$ bowlers,$10$ players are to be chosen for a team,which should include at least $4$ batsmen and at least $4$ bowlers. One batsman and one bowler,who are the captain and vice-captain of the team respectively,must be included. The total number of ways such a selection can be made is: