Let $A=\left[a_{i j}\right], a_{i j} \in Z \cap[0,4], 1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in(2,13)$ is $........$.

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 $......$

A company has $10$ employes. The company has decided to form a team including atleast three employes and also excluding atleast three employes. Then the number of ways of forming the team is

$m$ men and $n$ women are to be seated in a row so that no two women sit together. If $m > n$, then the number of ways in which they can be seated is

In a conference of $8$ persons, if each person shake hand with the other one only, then the total number of shake hands shall be

A total number of words which can be formed out of the letters $a,\;b,\;c,\;d,\;e,\;f$ taken $3$ together such that each word contains at least one vowel, is