Permutations an...

We number both the rows and the columns of an $$8$$ $$\times$$ $$8$$ chess-board with the numbers $$1$$ to $$8$$. A number of grains are placed onto each square, in such a way that the number of grains on a certain square equals the product of its row and column numbers. How many grains are there on the entire chessboard?

A hall has 12 gates. In how many ways can a man enter the hall through one gate and come out through a different gate?

$$\displaystyle \frac{^{\alpha}C_{2}}{^{\alpha + 1}C_{4}}$$ =

The total number of permutation of $$n$$ different things taken not more than $$r$$ at a time, where each thing may be repeated any number of times, is

The number of different ways in which five "alike dashes" and eight "alike dots" can be arranged using only seven of these "dashes" and "dots" is

There are three stations $$A,\space B$$ and $$C$$, five routes for going from station $$A$$ to station $$B$$ and four routes for going from station $$B$$ to station $$C$$.
Find the number of different ways through which a person can go from station $$A$$ to $$C$$ via $$B$$

Given six line segments of length $$2, 3, 4, 5, 6, 7$$ units. Then the number of triangles that can be formed by joining these lines is

Number of ways 6 rings can be worn on four fingers of one hand?

The value of $$\displaystyle ^{ 47 }{ { C }_{ 4 } }+\sum _{ r=1 }^{ 5 }{ ^{ 52-r }{ { C }_{ 3 } } } $$ is equal to

If $$m$$ parallel lines in a plane are intersected by a family of $$n$$ parallel lines, the number of parallelograms than can be formed is