# Problem of the MonthHint 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 $$\dbinom{n}{0}=1$$ 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 $$0\leq x_i\leq 9$$.

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.