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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 ?

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    $\begingroup$ 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 ... $\endgroup$
    – kimchi lover
    Commented Apr 10, 2023 at 10:46

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$\binom{2n}{3}$ 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.

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    $\begingroup$ Hi Surb!, simply answered, thank you ! the idea of using man and women excellent! $\endgroup$
    – Ben Cohen
    Commented Apr 10, 2023 at 10:51

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