Binomial Coefficient Formula

The binomial coefficient nk counts the number of ways to choose k items from n distinct items without regard to order.

The Formula

C(n,k)=n!k!(n−k)!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)=10C(5, 2) = 10 There are 10 ways to pick 2 items from 5, and 10 is the coefficient of a3b2a^3 b^2 in (a+b)5(a + b)^5.

Notation

(nk)\binom{n}{k} reads 'nn choose kk'; also written C(n,k)C(n, k) or nCk_nC_k

What This Formula Means

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

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

Formal View

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!} for 0≤k≤n0 \leq k \leq n; satisfies (nk)=(n−1k−1)+(n−1k)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} (Pascal's rule)

Worked Examples

Example 1

medium
Calculate (62)\binom{6}{2} using the formula (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}, and verify by listing all combinations of 2 items from {A,B,C,D,E,F}\{A, B, C, D, E, F\}.

Answer

(62)=15\binom{6}{2} = 15. Verified by listing all 15 two-item combinations.

First step

1
Apply formula: (62)=6!2!(6−2)!=6!2!⋅4!=6×5×4!2×1×4!=302=15\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

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Example 2

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

Example 3

medium
Calculate (83)\binom{8}{3} and explain what it counts.

Common Mistakes

  • Using (nk)\binom{n}{k} when order matters - if arrangement counts, use a permutation, which is k!k! times bigger.
  • Computing all of n!n! instead of canceling - (nk)=n!k!(n−k)!\binom{n}{k}=\frac{n!}{k!(n-k)!} cancels most factors; never multiply out giant factorials.
  • Forgetting (nk)=(nn−k)\binom{n}{k}=\binom{n}{n-k} - choosing which kk to keep equals choosing which n−kn-k to leave; use the smaller kk to compute.

Why This Formula Matters

The binomial coefficient is the bridge between counting and algebra: the same number that counts committees also appears in Pascal's triangle and the binomial theorem. Recognizing that 'how many ways to choose' and 'the coefficient in an expansion' are the SAME number is a genuine aha that unlocks both worlds. Recognizing it by "Am I counting selections of kk from nn where order doesn't matter (or the matching expansion coefficient)?" — rather than by familiar numbers — is what lets a student tell it apart from permutation and binomial distribution and factorial in a mixed problem set.

Frequently Asked Questions

What is the Binomial Coefficient formula?

The binomial coefficient (nk)\binom{n}{k} counts the number of ways to choose kk items from nn distinct items without regard to order. It equals n!k!(n−k)!\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?

(nk)\binom{n}{k} reads 'nn choose kk'; also written C(n,k)C(n, k) or nCk_nC_k

Why is the Binomial Coefficient formula important in Math?

The binomial coefficient is the bridge between counting and algebra: the same number that counts committees also appears in Pascal's triangle and the binomial theorem. Recognizing that 'how many ways to choose' and 'the coefficient in an expansion' are the SAME number is a genuine aha that unlocks both worlds. Recognizing it by "Am I counting selections of kk from nn where order doesn't matter (or the matching expansion coefficient)?" — rather than by familiar numbers — is what lets a student tell it apart from permutation and binomial distribution and factorial in a mixed problem set.

What do students get wrong about Binomial Coefficient?

The procedure for binomial coefficient is the easy part; the trap is using (nk)\binom{n}{k} when order matters. Asking "Am I counting selections of kk from nn where order doesn't matter (or the matching expansion coefficient)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Binomial Coefficient formula?

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

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