Help with Combinatorial argument ${2n\choose {2}} = 2{n\choose 2} + n^2$

Question

Show by a Combinatorics argument $${2n\choose {2}} = 2{n\choose 2} + n^2$$

I guess I'm having trouble with the "Combinatorial argument" part. Can anyone help out?

Answer 1

Suppose we have $2n$ flowers, number of ways to choose 2 of them is ${2n \choose 2}$ hence the left part of your equation

How else can we count them? we can part them in two n-groups of flowers, so we can choose 2 from the first group by ${n \choose 2}$ ways or 2 from the second group by ${n \choose 2}$ ways. Or we can choose one from the first group (n ways) and one from the second (again n ways) which gives us $n^2$ ways

Adding these together gives us 2${n \choose 2}$ + $n^2$ which is the right part of your equation

Answer 2

Hint: $\binom {2n}2$ counts ways to select $2$ items from a collection of $2n$ items

Hint: $2\binom n2+n^2 = \binom 2 1\binom n 2+\binom n1\binom n1$. What might this count?