How to choose these trainees?
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)?
The answer is 32.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Encoding to binary. Let s2 be singer, s1 be dancer and s0 be rapper. If each talent is distinct from the other, there are only 2^3-1 different talent sets. (000 not counted). Therefore each trainer can be mapped to a binary code
s2 s1 s0
0 0 1 : T1
0 1 0 : T2
0 1 1 : T3
1 0 0 : T4
1 0 1 : T5
1 1 0 : T6
1 1 1 : T7
Let us count the ways to fail. If none of the 3 trainees are rapper, then the only acceptable combination is (T2, T4, T6) If none are dancer, then (T1, T4, T5)and if none are singer, then(( T1, T2, T3) Therefore total acceptable combinations= no of possible selections of 3 objects from 7- 3failed combinations=7C3-3=32