C(n,k) = n!/(k!(n−k)!) counts ways to choose k items from n. Properties: C(n,0)=C(n,n)=1, C(n,k)=C(n,n−k), C(n,k)=C(n−1,k−1)+C(n−1,k). Binomial theorem: (a+b)ⁿ = ΣC(n,k)aⁿ⁻ᵏbᵏ. Pascal's triangle row n lists all C(n,k).
How to Use
Enter n and k:
- n: Total items
- k: Items chosen
- Result: C(n,k) and full row
Key Identities
- C(n,k) = C(n,n−k) (symmetry)
- ΣC(n,k) = 2ⁿ (sum of row)
- C(n,k) = C(n−1,k−1)+C(n−1,k)
Binomial Theorem
(x+y)ⁿ = Σ C(n,k) xⁿ⁻ᵏyᵏ. The coefficients are exactly Pascal's triangle row n.
Step-by-Step Instructions
- 1Enter n (total).
- 2Enter k (chosen).
- 3View C(n,k).
- 4See Pascal's triangle row.
- 5Check identities.