Math and Logic

A Test Question

Today, Pearl’s 99 grandchildren are coming to visit! She loves to spoil them, so she opens her purse and finds 1313 dollar bills.

In how many different ways can Pearl distribute those dollars amongst her grandchildren? Keep reading to find out, or skip to today’s challenge for a similar problem.

As we’ll see, there are a lot of ways for Pearl to distribute her dollars! So, let’s start with a smaller example. Last week, Pearl’s 33 favorite grandchildren visited, and at that time, she had 44 dollar bills to give them. To visualize how they could be distributed, she laid them out in a row, along with some pencils to divide them into 33 groups.

We’ll represent the dollars with stars \large \star and divisions between groups with bars .\large{|}. One arrangement that Pearl found was        \large \star \; | \, \star \star \; | \; \star which represents 11 dollar for the first grandchild, 22 dollars for the second, and 11 dollar for the third. Another arrangement was    \large \star \; | \: | \, \star \star \, \star which represents 11 dollar for the first grandchild, 00 dollars for the second, and 33 dollars for the third.

To create 33 groups, we need 22 bars to separate the stars. So, to count the total number of arrangements into groups, we can count where in the line of stars and bars we can place those bars to define the groups.

With 44 stars and 22 bars, there are 66 positions in the line. Generally, if we want to choose kk things from a set of nn things, there are (nk)=n!k!(nk)! \binom n k = \frac{n!}{k!(n-k)!} ways to do so. So, by choosing which positions the 22 bars take, we see there are (62)=6!2!(62)!=15 \binom 6 2 = \frac{6!}{2!(6-2)!} = 15 ways to create 33 groups of stars. That means Pearl had 1515 different ways to distribute 44 dollar bills to her 33 favorite grandchildren.

Now we can use this same method to count the number of ways for Pearl to distribute 1313 dollar bills amongst all 99 of her grandchildren — we represent it with 1313 stars in a row, and we can divide them into 99 groups using 88 bars. Then there are 13+8=2113+8=21 positions available for the bars, and the rest must be filled with stars. That makes (218)=203940 \binom{21}{8} = 203940 arrangements, or possible distributions of dollars!

Woah… hopefully Pearl has a way to narrow down the choices! Now, can you use stars and bars to solve today’s challenge?

Today's Challenge

Leticia is writing a test question for her math students. She wants them to find solutions to the equation x+y+z=12, x + y + z = 12, where x,y,x, y, and zz must all be non-negative integers.

If she changes the "non-negative" condition to "positive," then the number of solutions will decrease. By how many? \\[-0.3em]

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