1. ## Combinatorics

1. In how many ways can two squares be selected from an 8-by-8 chessboard so that they are not in the same row or the same column.

2. In how many ways can we place r red balls and w white balls in n boxes so that each box contains at least one ball of each colour.

3. Find a formula for $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+...+\ binom{n}{n}^2$

Thanks

2. 1. In how many ways can two squares be selected from an 8-by-8 chessboard so that they are not in the same row or the same column.
See this topic.

2. In how many ways can we place r red balls and w white balls in n boxes so that each box contains at least one ball of each colour.
Put one ball of each color in each box and place the rest of the balls at random (i.e., in all possible ways).

3. Find a formula for
Consider $(1+x)^n(1+x)^n=(1+x)^{2n}$. Write the coefficient of $x^n$ in both sides and use the fact that ${n\choose k}={n\choose n-k}$. See also this Wikipedia page, (8).

In the future, please do not post bare questions. People would be more than willing to give you hints and show direction if you present your experience with the problem: what you have tried, what worked and what didn't, what exactly your difficulty is.

3. Thanks for that.

They are not bare, I don't know what to do at all.

4. Originally Posted by usagi_killer
I don't know what to do at all.
If that is true, then why are you being asked to solve these problems?
Don't you have any textbook or lecture notes?
You must have given examples of these.