A student dance committee is to be formed consisting of 2 boys and 4 girls. If the membership is to be chosen from 5 boys and 6 girls, how many different committees are possible?

150 different committees are possibleSolution:Given that a student dance committee is to be formed consisting of 2 boys and 4 girlsThe membership is to be chosen from 5 boys and 6 girlsTo find : number of different possible committeesA combination is a selection of all or part of a set of objects, without regard to the order in which objects are selectedThe formula for combination is given as:[tex]n C_{r}=\frac{n !}{(n-r) ! r !}[/tex]where "n" represents the total number of items, and "r" represents the number of items being chosen at a timeWe have to select 2 boys from 5 boysSo here n = 5 and r = 2[tex]\begin{aligned} 5 C_{2} &=\frac{5 !}{(5-2) ! 2 !}=\frac{5 !}{3 ! 2 !} \\\\ 5 C_{2} &=\frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} \\\\ 5 C_{2} &=10 \end{aligned}[/tex]We have to select 4 girls from 6 girlsHere n = 6 and r = 4[tex]\begin{aligned} 6 C_{4} &=\frac{6 !}{(6-4) ! 4 !}=\frac{6 !}{2 ! 4 !} \\\\ 6 C_{4} &=\frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 4 \times 3 \times 2 \times 1}=15 \end{aligned}[/tex]Committee is to be formed consisting of 2 boys and 4 girls:So we have to multiply [tex]5 C_{2}[/tex] and [tex]6 C_{4}[/tex][tex]5 C_{2} \times 6 C_{4}=10 \times 15=150[/tex]So 150 different committees are possible