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 binomial coefficient \binom{n}{k} counts the number of ways to choose k items from n distinct items without regard to order. It equals \frac{n!}{k!(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).

Example 3

medium
Calculate \binom{8}{3} and explain what it counts.

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 binomial coefficient \binom{n}{k} counts the number of ways to choose k items from n distinct items without regard to order. It equals \frac{n!}{k!(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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