How many ways can the agency choose three trainees to form a group, provided that the group must have at least..........

Question

An entertainment agency has seven trainees. Each of the trainees does at least one of dancing, singing, and rapping, and no two trainees have the same skill set.
How many ways can the agency choose three trainees to form a group, provided that the group must have at least one dancer, one singer, and one rapper (who are not necessarily distinct)?

Answer 1

32 ways

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Dancing- d, singing- s, rapping- r
Skill set as per trainee, 7 possible options as:
1- dsr
2- ds
3- dr
4- sr
5- d
6- s
7- r

Options of 3 trainees with required skill test:
1. dsr+ any 2 others --> 15 ways
2. ds+ 2 others with r included in any --> 4+3+1+1= 9 ways
3. dr+ 2 others with s included in any --> 3+2= 5 ways
4. sr+ 2 others with d included in any --> 2 ways
5. d+s+r --> 1 way

Total- 15+9+5+2+1= 32 ways