A group of 7 members having a majority of boys is to be formed out of 6 boys…
2012
A group of 7 members having a majority of boys is to be formed out of 6 boys and 4 girls. The number of ways the group can be formed is
Answer: B. 100 — ConceptFor an unordered selection, choosing r people from n distinct people gives C(n,r) ways. When categories and a condition are involved, list every…
- A.
80
- B.
100
- C.
90
- D.
110
Attempted by 2 students.
Show answer & explanation
Correct answer: B
Concept
For an unordered selection, choosing r people from n distinct people gives C(n,r) ways. When categories and a condition are involved, list every disjoint valid category split, count each split with a product of combinations, and then add the counts.
Application
With 7 members and a majority of boys, the possible boy–girl splits are (4,3), (5,2), and (6,1).
For (4,3), the count is C(6,4) × C(4,3) = 15 × 4 = 60.
For (5,2), the count is C(6,5) × C(4,2) = 6 × 6 = 36.
For (6,1), the count is C(6,6) × C(4,1) = 1 × 4 = 4.
Adding the disjoint cases gives 60 + 36 + 4 = 100 ways.
Cross-check
There are C(10,7) = 120 unrestricted groups. A group without a boy majority has either 3 boys and 4 girls, counted by C(6,3) × C(4,4) = 20, or fewer boys, which is impossible because only 4 girls are available. Thus 120 − 20 = 100.