Problem of the Month
Hint for Problem 5: February 2024

First, some hints on the exercises.

1. List the two-element subsets of $\{1,2,3,4,5\}$.

2. The answers are $1$ and $n$. It might take a moment of reflection to convince yourself that ${\displaystyle (\genfrac{}{}{0ex}{}{n}{0})}=1$ makes sense.

3. If $k$ objects are chosen from a set of $n$ objects, then how many objects are not chosen?

4. If you are to choose $k+1$ integers from the set $\{1,2,3,\dots ,n,n+1\}$, then either $n+1$ is chosen or it is not.

5. The quantity $(1+x{)}^n$ is equal to the product of $n$ copies of $(1+x)$. Try expanding $(1+x{)}^n$ for a few small values of $n$ without collecting like terms. As an example, think about how an $x^3$ term could arise during the expansion of $(1+x)(1+x)(1+x)(1+x)(1+x)$.

Below are the hints for the main problems.

1. If you have never seen a proof of (a)(i), try writing the sum $1+2+3+\cdots +n$ in reverse order directly under the sum $1+2+3+\cdots +n$. Now add each column. For (a)(ii), consider the possible values of $x$. For (a)(iii), consider the equation $x+y+z=n-1$ for a fixed pair $(x,y)$.

2. Imagine arranging the $n$ identical balls in a row and placing $r-1$ sticks between them. By doing this, you have partitioned the $n$ balls into $r$ smaller groups.

3. Introduce a new variable, $x_0$, and consider the equation $x_0+x_1+\cdots +x_r=n$.

4. The non-negative integers $x$ with $x<{10}^{10}$ are exactly the integers that have at most $10$ digits. Consider the equation $x_1+x_2+\cdots +x_{10}=21$ where $0\le x_i\le 9$.

   There was an omission in part (d). The original question said "integers" where it should have said "non-negative integers".

5. This question can be analyzed by examining the equation $x_1-x_2+x_3-x_4+x_5-x_6+x_7-x_8=0$. Rearrange this equation and use the ideas from (b) and (d).

6. Let $x=a-1$, $y=b-1$, and $z=c-1$. Find the number of non-negative integer solutions to $x+y+z=2024$ with $x$, $y$, and $z$ distinct.