A class consists of 15 boys and 17 girls.
- If two students have to be in the same group, how many groups of 12 students can be formed from this class?
- If one boy A and one girl B cannot be in the same committee, how many ways can a committee consisting of 4 boys and 4 girls be chosen from the class?
Now for 1, I reason the following,
${^{30}C_{10}} \times {^2C_2} + {^{30}C_{12}}\ $
Where the first expression accounts for the pair of students in the group, and the second, where they are not. In my mind these are the only two possibilities.
For 2,
${^{14}C_{4}} \times {^{16}C_{3}} \times {^{1}C_{1}} + {^{14}C_{3}} \times {^{1}C_{1}} \times {^{16}C_{4}} + {^{14}C_{4}} \times {^{16}C_{4}} \ $
Where the first term is where A is not present, the second where B isn't, and the third where neither is present.
Is this the correct approach for solving this kind of question?