We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen too.
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We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen too. In how many ways can the selection be made?
Show answer & explanation
We break this into two cases based on whether person A is selected. Case 1: A is not selected. We choose 6 from the remaining 7 persons: C(7,6) = 7 ways. Case 2: A is selected, so B must also be selected. We need to choose 4 more from the remaining 6 persons: C(6,4) = 15 ways. Total: 7 + 15 = 22 ways.
Step-by-step Derivation:
Case Analysis:
Total persons: 8 (including A and B)
Need to select: 6 persons
Constraint: If A is chosen, then B must be chosen.
Case 1: Person A is NOT chosen
- We need to select 6 from the remaining 7 persons (everyone except A)
- B may or may not be chosen (no constraint applies)
- Number of ways = C(7, 6) = 7
Case 2: Person A IS chosen
- B MUST be chosen (due to constraint)
- We have already selected A and B (2 persons)
- We need to select 4 more from the remaining 6 persons (8 - 2 = 6)
- Number of ways = C(6, 4) = (6 × 5 × 4 × 3) / (4 × 3 × 2 × 1) = 360 / 24 = 15
Total ways = Case 1 + Case 2 = 7 + 15 = 22
Note: The provided answer key shows option B = 21, but the correct mathematical answer is 22 (option D). There may be a transcription error in the original assessment.