Binomial Coefficient Formula

The Formula

C(n, k) = \frac{n!}{k!(n - k)!}

When to use: Same as combination count, but now viewed as a coefficient in algebraic expansions.

Quick Example

C(5, 2) = 10 There are 10 ways to pick 2 items from 5, and 10 is the coefficient of a^3 b^2 in (a + b)^5.

Notation

\binom{n}{k} reads 'n choose k'; also written C(n, k) or _nC_k

What This Formula Means

The number of ways to choose k items from n items, written C(n, k) or \binom{n}{k}.

Same as combination count, but now viewed as a coefficient in algebraic expansions.

Formal View

\binom{n}{k} = \frac{n!}{k!(n-k)!} for 0 \leq k \leq n; satisfies \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} (Pascal's rule)

Worked Examples

Example 1

medium
Calculate \binom{6}{2} using the formula \binom{n}{k} = \frac{n!}{k!(n-k)!}, and verify by listing all combinations of 2 items from \{A, B, C, D, E, F\}.

Solution

  1. 1
    Apply formula: \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} = \frac{30}{2} = 15
  2. 2
    List all 2-item combinations from \{A,B,C,D,E,F\}: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF
  3. 3
    Count: 15 combinations ✓
  4. 4
    Confirms formula gives the correct count

Answer

\binom{6}{2} = 15. Verified by listing all 15 two-item combinations.
The binomial coefficient \binom{n}{k} counts the number of ways to choose k items from n without regard to order. Order doesn't matter in combinations (unlike permutations). The formula cancels repeated arrangements via the k! in the denominator.

Example 2

hard
A fair coin is flipped 5 times. Using P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, find P(X=3) (exactly 3 heads).

Common Mistakes

  • Swapping n and k in the formula — C(5, 2) \neq C(2, 5); k cannot exceed n
  • Forgetting that C(n, 0) = 1 and C(n, n) = 1 — there is exactly one way to choose nothing or everything
  • Computing \frac{n!}{k!} instead of \frac{n!}{k!(n-k)!} — omitting the (n-k)! in the denominator

Why This Formula Matters

Appears in the binomial theorem, probability distributions, and Pascal's triangle.

Frequently Asked Questions

What is the Binomial Coefficient formula?

The number of ways to choose k items from n items, written C(n, k) or \binom{n}{k}.

How do you use the Binomial Coefficient formula?

Same as combination count, but now viewed as a coefficient in algebraic expansions.

What do the symbols mean in the Binomial Coefficient formula?

\binom{n}{k} reads 'n choose k'; also written C(n, k) or _nC_k

Why is the Binomial Coefficient formula important in Math?

Appears in the binomial theorem, probability distributions, and Pascal's triangle.

What do students get wrong about Binomial Coefficient?

C(n, k) = C(n, n - k). Choosing k to include is the same as choosing n - k to exclude.

What should I learn before the Binomial Coefficient formula?

Before studying the Binomial Coefficient formula, you should understand: combination, factorial.

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