Total number of four-digit odd numbers that can be formed using $0, 1, 2, 3, 5, 7$ (repetition is allowed) is:

How many numbers between $5000$ and $10000$ can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ if no digit is repeated?

The total number of words (with or without meaning) that can be formed out of the letters of the word $'DISTRIBUTION'$ taken four at a time,is equal to...............

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is:

If $n, r$ are two positive integers such that $1 \leq r < n$,then ${ }^{n} P_{r+1} + r^2 { }^{n-1} P_{r-1} + (r+1) { }^{n-1} P_{r} + r { }^{n-1} P_{r-1} =$

$10$ men and $6$ women are to be seated in a row so that no two women sit together. The number of ways they can be seated is: