In how many ways can the sample be selected if it must have at least 2 male and 1 female mice?

Question

A sample of 5 mice is to be chosen from 7 male and 6 female mice. In how many ways can the sample be selected if it must have at least 2 male and 1 female mice?

I tried doing 7C2 * 6C1 but it seems to be wrong as the answer is 1155 but I got 126

Answer 1

For a selection to meet the conditions, it must be exactly one of the following types (as may be seen by considering what genders the "optional" mice may have):

• 4 male 1 female: $\binom74\binom61=210$ ways
• 3 male 2 female: $\binom73\binom62=525$ ways
• 2 male 3 female: $\binom72\binom63=420$ ways

The number of possible selections is $210+525+420=1155$.