Combinatorial Argument

Question

Can you give a Combinatorial argument that $$\binom{3n}{3}=3\binom{n}{3}+(3)(2)\binom{n}{2}\binom{n}{1}+\binom{n}{1}\binom{n}{1}\binom{n}{1}?$$

Answer 1

You have three urns, each with $n$ objects in it. The number of ways to select three objects total from the urns is the left hand side ($ ({3n\atop3})$) .

The right hand side consists of three terms: the first term ($3({n\atop3})$) is the number of ways to select the three objects with all three in the same urn. The second term ($3\cdot2\cdot({n\atop2})({n\atop1})$) is the number of ways to select the three objects by first picking an urn and selecting two objects, and then selecting one of the two remaining urns and selecting an object. The last term ($({n\atop1})({n\atop1})({n\atop1})$) is the number of ways to select the three objects with one selected from each urn. Adding these together gives you the total number of ways to select three objects from the urns.