The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4,$ and $5$ in which at least two digits are identical is

$^nC_r + ^{n-1}C_r + ...... + ^rC_r$ =

From a group of $7$ men and $6$ women,five persons are to be selected to form a committee so that at least $3$ men are there in the committee. In how many ways can it be done?

How many numbers are there between $100$ and $1000$ in which all the digits are distinct?

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible,if the first three questions have $4$ choices each and the next three have $5$ choices each?

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