For today, we will discuss the problem given on February 8, 2017:
Problem: Find the number of quadruples $(a, b, c, d)$ of integers satisfying $a + b + c + d = 100, \ a \geq 23, b > 18, c \geq 11, d \geq 1.$
Solution: Put $a' = a-22, b' = b - 18, c' = c - 10.$ Then we want the number of positive integer solutions to $a' + 22 + b' + 18 + c' + 10 + d = 100,$ or $a' + b' + c' + d = 50.$ This is a Stars & Bars problem, with answer
$\binom{49}{3} = 18424.$
This question could be misinterpreted as a number theory problem. Because of this, students may try to use number theoretic approaches to try and answer the problem. This is a disguised Stars and Bars problem.
We begin with $100$ stars and $4$ bins, where each bin represents a variable. Reserve $23$ stars for the first bin representing $a$, $19$ for the second bin representing $b$, $11$ for the third bin representing $c$, and $1$ for the last bin representing $d$. A common mistake that students could make is to unnecessarily account for $b = 18$ so instead of reserving $19$ in the second bin, they would reserve $18$.
After the stars are reserved, there are $100 - (23 + 19 + 11 + 1) = 46$ remaining stars freely able to be distributed however you want. We will use $3$ bars to indicate the bins as spaces between the bars and arrange the $46 + 3 = 49$ objects in $\binom{49}{3} = 18424$ ways.
If you have any approach that you would like to share, please post below! Let us know your thoughts below!