Problem
COM-B1-M03-P014 Choosing from Two Groups
#14
★★★☆☆ Level 3 of 5
Prove that the number of ways to choose \(3\) people from \(m\) boys and \(n\) girls is \(\binom{m}{3}+\binom{m}{2}\binom{n}{1}+\binom{m}{1}\binom{n}{2}+\binom{n}{3}\).
Split by the number of boys in the team.
A team of \(3\) can contain \(3,2,1,0\) boys. These cases are disjoint. In each case choose the required number of boys and girls, giving the stated sum.
This is a special case of Vandermonde's identity.