If a number is chosen at random from the four-digit numbers formed by using the digits $0, 1, 2, 3, 4, 6$ without repetition,then the probability that it is divisible by $4$ is

The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

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

Group $A$ consists of $7$ boys and $3$ girls,while group $B$ consists of $6$ boys and $5$ girls. The number of ways,$4$ boys and $4$ girls can be invited for a picnic if $5$ of them must be from group $A$ and the remaining $3$ from group $B$,is equal to:

Among the inequalities below,which ones are true for all natural numbers $n > 1000$?
$I. n! \leq n^n$
$II. (n!)^2 \leq n^n$
$III. 10^n \leq n!$
$IV. n^n \leq (2n)!$

If all the numbers which are greater than $6000$ and less than $10000$ are formed with the digits $3, 5, 6, 7, 8$ without repetition of the digits,then the difference between the number of odd numbers and the number of even numbers among them is