A class of children contains 14 boys and 12 girls. How many ways are there to choose a team of six children from the class if it must contain at least two boys and at least two girls?
I already know the answer(191191), which was found by considering the cases of 2 boys + 4 girls, 3 boys + 3 girls and, 4 boys and 2 girls separately - then adding up the numbers.
However I am not sure why I can't do the following, so would like the error to be pointed out:
First choose 2 boys, then choose two girls, then choose any two pupils from the remaining 22. So
2 boys = 14C2 combinations = 91
2 girls = 12C2 combinations = 66
2 remaining pupils = 22C2 combinations = 231
Total number = 91*66*231 = 1387386