Lesson 9 of 11 · 7 min
Selection with conditions
NCERT §6.4
The sports committee needs a team of 5 from 4 girls and 6 boys, 10 students in all. The rules may ask for exactly 2 girls, at least 2 girls, or insist that the captain plays.
The lesson in notes
In short
'Exactly' means one case; 'at least' and 'at most' usually mean adding several cases, or subtracting the unwanted cases from the total.
A team of 5 from 4 girls and 6 boys with exactly 2 girls: 4C2 × 6C3 = 6 × 20 = 120.
The same team with at least 2 girls, by cases: 2 girls 120, 3 girls 4C3 × 6C2 = 60, 4 girls 4C4 × 6C1 = 6, total 186.
By subtraction: all teams 10C5 = 252; no girl 6C5 = 6; one girl 4 × 6C4 = 60. So 252 − 6 − 60 = 186, the same answer.
A particular person always included: put that person in and choose the rest. A 5-member team from 10 that must include the captain: 9C4 = 126. Teams that leave the captain out: 9C5 = 126. Together 252 = 10C5.