If a man and his wife enter a bus in which five seats are vacant,then the number of different ways in which they can be seated is

$A$ committee of $11$ members is to be formed from $8$ men and $5$ women. If $m$ is the number of ways to form the committee with at least $6$ men and $n$ is the number of ways to form the committee with at least $3$ women,then:

The number of arrangements of the letters of the word $SATAYPAUL$ such that no two $A$ are together and the middle letter is a consonant,is

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded and $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is

In how many ways can a committee consisting of $5$ men and $6$ women be formed from $8$ men and $10$ women?

How many numbers consisting of $5$ digits can be formed in which the digits $3, 4$ and $7$ are used only once and the digit $5$ is used twice?