The number of matrices of order $3 \times 3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is$....$

How many numbers between $5000$ and $10,000$ can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ each digit appearing not more than once in each number

It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?

If for some $\mathrm{m}, \mathrm{n} ;{ }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3$ and ${ }^{n-1} P_3:{ }^n P_4=1: 8$, then ${ }^n P_{m+1}+{ }^{n+1} C_m$ is equal to

How many numbers of $6$ digits can be formed from the digits of the number $112233$

There are $5$ students in class $10,6$ students in class $11$ and $8$ students in class $12.$ If the number of ways, in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100 \mathrm{k}$, then $\mathrm{k}$ is equal to $......$