Combinatorial proof $\binom{2n}{3} = 2 \binom{n}{3} + 2n \binom{n}{2}$

Give a Combinatorical Proof for the identity $$\binom{2n}{3} = 2 \binom{n}{3} + 2n \binom{n}{2}$$

The LHS is pretty easy binomial(2n, 3) represents the number of ways to choose a subset of 3 elements from a set of 2n elements. I'm struggling with the RHS to find a combinatorial story , I tried to divide the problem into two different problems, any suggestions ?

• Divide your $2n$ items into 2 groups with $n$ items apiece. You can pick your 3 items from the first group, or from the second group, or one from the first and 2 from the second, or ... Commented Apr 10, 2023 at 10:46

1 Answer

$$\binom{2n}{3}$$ is a the number of group with $$3$$ animals among a group with $$n$$ males and $$n$$ females. But to do such group, you can also take either $$3$$ males, or 3 females, or 1 male and 2 females, or 2 males and 1 female, which gives $$2\binom{n}{3}+2\binom{n}{1}\binom{n}{2},$$ possibilities.

• Hi Surb!, simply answered, thank you ! the idea of using man and women excellent! Commented Apr 10, 2023 at 10:51