• In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find $N$.

In the following $(8\times 5)$ grid, how many shortest routes are there from point $A$ to point $B$?

In the following $5\times 4\times 3$ grid system, how many shortest routes are there from point $A$ to point $B$?

In the following $5\times 4\times 3$ grid, how many shortest routes are there from point $A$ to point $B$ on its surface?

There are $5$ red balls and $4$ green balls in a bag. One ball is retrieved a time until all the balls are taken out. How many possible ways are there such that all the red balls are taken out before all the green balls are taken out?

There are $5$ red balls, $4$ green balls and $3$ yellow balls in a bag. One ball is retrieved a time until all the balls are taken out. What is the probability that all the red balls are retrieved before all the green or yellow balls are retrieved?

Show that for any positive integer $n$, the value of $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer.

How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)