Binomial Coefficient: n Choose k
C(n,k) counts the number of ways to choose k items from n items without regard to order. It equals n!/(k!(n-k)!) and appears in Pascal's triangle and the binomial theorem (x+y)^n.
Why This Mathematical Concept Matters
Why: Binomial coefficients count combinations and appear in probability (binomial distribution), algebra (expansion of (x+y)^n), and combinatorics. Pascal's triangle row n gives coefficients for (a+b)^n.
How: Compute n!/(k!(n-k)!). For large n, use Pascal's identity or built-in functions. C(n,0)=C(n,n)=1; C(n,1)=C(n,n-1)=n.
- โPascal's triangle: each entry is sum of two above it.
- โC(n,k) is the coefficient of x^k in (1+x)^n.
- โSum of row n in Pascal's triangle = 2^n.
๐ Examples โ Click to Load
Enter Values
Pascal Row Chart
Coefficient Distribution
๐ Step-by-Step Breakdown
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
C(n,k) = number of k-element subsets of an n-element set.
Pascal's triangle: row n contains C(n,0), C(n,1), ..., C(n,n).
๐ Key Takeaways
- โข C(n,k) = n! / (k!(n-k)!) โ number of ways to choose k from n
- โข Pascal's triangle: each entry is sum of two above โ C(n,k) = C(n-1,k-1) + C(n-1,k)
- โข Binomial theorem: (x+y)^n = ฮฃ C(n,r) x^(n-r) y^r for r = 0 to n
- โข Symmetry: C(n,k) = C(n, n-k)
- โข Row sum: C(n,0) + C(n,1) + โฆ + C(n,n) = 2^n
๐ก Did You Know?
๐ How It Works
The binomial coefficient C(n,k) counts combinations: ways to choose k items from n when order doesn't matter. For expansion, (x+y)^n = ฮฃ C(n,r) x^(n-r) y^r โ each term has coefficient C(n,r) with powers determined by r.
Pascal's triangle builds row by row: each number is the sum of the two above it.
๐ Worked Example: (x+y)^3
Step 1: (x+y)^3 = C(3,0)xยณyโฐ + C(3,1)xยฒyยน + C(3,2)xยนyยฒ + C(3,3)xโฐyยณ
Step 2: C(3,0)=1, C(3,1)=3, C(3,2)=3, C(3,3)=1
Result: (x+y)ยณ = xยณ + 3xยฒy + 3xyยฒ + yยณ
Pascal row 3: 1, 3, 3, 1
๐ Real-World Applications
๐ Binomial Distribution
C(n,k) p^k (1-p)^(n-k) for k successes in n trials.
๐ฌ Genetics
Mendelian inheritance โ 3:1 ratio from (1+1)^2 expansion.
๐ Finance
Binomial option pricing model.
๐ฒ Probability
Coin flips, dice, Bernoulli trials.
๐ Algebra
Expanding (a+b)^n, polynomial identities.
๐ป Algorithms
Combinatorial counting in CS.
โ ๏ธ Common Mistakes to Avoid
- k > n: C(n,k) = 0 when k > n. Check bounds.
- Forgetting symmetry: Use C(n,n-k) when k > n/2 to avoid overflow.
- Wrong expansion order: (x+y)^n: first term x^n, last term y^n; powers sum to n.
- Factorial overflow: For large n, use multiplicative formula, not n! directly.
- Row sum check: Always verify ฮฃ C(n,r) = 2^n.
๐ฏ Expert Tips
๐ก Use C(n,n-k)
When k > n/2, compute C(n,n-k) โ fewer multiplications, same result.
๐ก Multiplicative Formula
C(n,k) = nร(n-1)รโฆร(n-k+1) / k! โ avoids n! overflow.
๐ก Pascal Row
Row n gives C(n,0) through C(n,n). Sum = 2^n.
๐ก Verify Expansion
Substitute x=y=1: (1+1)^n = 2^n = sum of coefficients.
๐ Reference Table
| Identity | Formula |
|---|---|
| Symmetry | C(n,k) = C(n, n-k) |
| Pascal | C(n,k) = C(n-1,k-1) + C(n-1,k) |
| Row sum | ฮฃ C(n,r) = 2^n |
| C(n,0) | 1 |
| C(n,n) | 1 |
| Binomial | (a+b)^n = ฮฃ C(n,r) a^(n-r) b^r |
๐ Quick Reference
๐ Practice Problems
โ FAQ
What is C(n,k)?
Number of ways to choose k items from n, order doesn't matter. Same as "n choose k".
Why C(n,0)=1?
One way to choose nothing: the empty set.
What is Pascal's triangle?
Triangular array where each number is the sum of the two above. Row n gives C(n,0) to C(n,n).
How does binomial theorem work?
(x+y)^n expands to sum of C(n,r)รx^(n-r)รy^r for r=0 to n.
When is C(n,k) largest?
When k is near n/2. For even n, k=n/2. For odd n, k=โn/2โ or โn/2โ.
Can k be negative?
No. C(n,k)=0 when k < 0 or k > n.
Relation to combination?
C(n,k) is the same as the combination C(n,r). Both denote "n choose k".
๐ Summary
C(n,k) counts combinations and appears in Pascal's triangle and the binomial theorem. Use symmetry C(n,k)=C(n,n-k) for efficiency. Row n sums to 2^n. (x+y)^n expands using these coefficients.
โ Verification Tip
Verify: sum of row n = 2^n. For expansion, set x=y=1: (1+1)^n = 2^n = sum of coefficients.
๐ Next Steps
Explore the Combination Calculator, Permutation Calculator, or Factorial Calculator for related combinatorics.
โ ๏ธ Disclaimer: Results for educational use. For large n, overflow may occur. Verify critical calculations independently.