Fix the mistake: Counting Cases ② (Combinations)
Combinations① Definition and Calculation of nCr
The total number of ways to simply choose r objects from n objects (without considering order) is called a combination, nCr.
Combination Formula
nCr = n! / (r!(n−r)!)、 nCr = nC(n−r)
Example①
Choosing 3 committee members from 10 people: 10C3 = (10×9×8)/(3×2×1) = 120 ways
Choosing 3 committee members from 10 people: 10C3 = (10×9×8)/(3×2×1) = 120 ways
Example② (Application of Combinations)
Choosing 3 men and 2 women from 6 men and 4 women: 6C3 × 4C2 = 20 × 6 = 120 ways
Choosing 3 men and 2 women from 6 men and 4 women: 6C3 × 4C2 = 20 × 6 = 120 ways
Key Points
- Order does not matter → nCr
- nCr = nC(n−r) (complementary relationship)
- Lines determined by n points on a plane: nC2, triangles: nC3
Practice Problems
- Calculate 6C2 and 10C4.
- There are 8 points on a plane, with no 3 points collinear. How many triangles can be formed?
- How many ways can 5 committee members be chosen from 12 people?
Answers
- 6C2=15、10C4=210
- 8C3=56 triangles
- 12C5=792 ways
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