I am struggling on how to approach combinatoric proof problems. The identity to prove is $$\binom{10}{5} = \binom{8}{3} + \binom{8}{4} +\binom{8}{4} + \binom{8}{5}$$
I can prove this algebraically but the question says prove combinatorially. I also could use pascals identity to solve the RHS to the LHS.
How would I describe this combinatorially? Any guidance would be greatly appreciated.
On
There are a few standard combinatorial interpretations of $\binom nk$. Let's try one: suppose $\binom{10}5$ refers to the number of ways to pick $5$ objects from among $10$.
The right side speaks of picking from among $8$. To match the left side, let's assume we have $8$ objects plus $2$ special objects, and we're looking for the number of ways to pick a total of $5$ from among that group. Can you see how you get the expression on the right?
Think of $\binom{10}{5}$ as the number of ways to choose a team of $5$ children from a group of $10$.
Now suppose we have a group of $8$ girls and $2$ boys, Matthew and Paul. $\binom{8}{5}$ counts how many teams have $5$ girls. $\binom{8}{4}$ counts how many teams have $4$ girls and Matthew. $\binom{8}{4}$ counts how many teams have $4$ girls and Paul. $\binom{8}{3}$ counts how many teams have $3$ girls and both Matthew and Paul.