The number of ways to seat $3$ men and $2$ women in a bus such that the total number of seated men and women on each side is $3$ is:

The numbers $1, 2, 3,$ and $4$ are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. $A$ person draws two slips from the box,one after the other,without replacement. Describe the sample space for the experiment.

Using the digits $0, 2, 4, 6, 8$ not more than once in any number,the number of $5$-digit numbers that can be formed is:

Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arrangements do all the vowels always occur together?

We are to form different words with the letters of the word $INTEGER$. Let $m_1$ be the number of words in which $I$ and $N$ are never together and $m_2$ be the number of words which begin with $I$ and end with $R$,then $m_1/m_2$ is equal to

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is