A committee of 5 members is to be formed from a group of 10 people.
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A committee of 5 members is to be formed from a group of 10 people. In how many ways can the committee be formed if 2 specific people refuse to serve together?
Show answer & explanation
Use complementary counting: Total committees without restriction minus committees with both conflicting members. Total = C(10,5) = 252. Committees with both conflicting members = C(8,3) = 56 (select 3 more from remaining 8). Valid committees = 252 - 56 = 196.
Step-by-step Derivation:
Step 1: Calculate total committees without restriction.
C(10,5) = 10!/(5!×5!) = (10×9×8×7×6)/(5×4×3×2×1) = 30240/120 = 252
Step 2: Calculate committees where both conflicting members serve together.
If both specific people are included, we need to select 3 more members from the remaining 8 people.
C(8,3) = 8!/(3!×5!) = (8×7×6)/(3×2×1) = 336/6 = 56
Step 3: Apply complementary counting.
Valid committees = Total - (Both conflicting members together)
Valid committees = 252 - 56 = 196