In a party there are $r$ boys and $m$ girls. The first boy dances with $5$ girls, the second boy dances with $6$ girls, and the last boy dances with all girls. What is the relation between $r$ and $m$? I managed to find that $m=r+4,$ but it was not using combinatorics.
Let me explain how I found that answer: first boy dances with $a_{1} $ girls, second boy dances with $a_{2} $girls, last boy dances with $a_{r}$ girls. Then $5=a_{1}=1+4, 6=a_{2}=2+4,\cdots, m=a_{r}=4+r.$
My question: How does one solve this question by using combinatorics?