Combinatorics / Inclusion/Exclusion Question

Quote:

Originally Posted by floorplay

Hi everyone, first time posting on this forum and was hoping if someone would be willing to explain how to do the below question. Any help appreciated, thanks.

The question is to find the number of solutions to $\sum\limits_{k = 1}^5 {u_k } = 24,\quad 4 \le u_1 \;\& \;u_2 < 7$
Let $S_k$ be the event that $u_1=k$ .
Let $T_k$ be the event that $u_2=k$ .
Let $\|S_k\|$ be the number of solutions in $S_k$ .

The answer to the posed question is $\sum\limits_{k = 4}^6 {\sum\limits_{j = 4}^6 {\left\| {S_k \cap T_j } \right\|} }$ ,
where $\left\| {S_k \cap T_j } \right\|=\binom{26-k-j}{2}$ .