Binomial Coefficient

Probability
definition

Also known as: n choose k, choose function, C(n,k)

Grade 9-12

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The binomial coefficient \binom{n}{k} counts the number of ways to choose k items from n distinct items without regard to order. Appears in the binomial theorem, probability distributions, and Pascal's triangle.

Definition

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)!}.

💡 Intuition

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

🎯 Core Idea

The binomial coefficient bridges counting (combinations) and algebra (polynomial expansion).

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.

Formula

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

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Use Pascal's triangle for small values: each entry is the sum of the two entries above it. For calculation, cancel common factors before multiplying to keep numbers manageable.

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)

🚧 Common Stuck Point

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

⚠️ 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

Frequently Asked Questions

What is Binomial Coefficient in Math?

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)!}.

What is the Binomial Coefficient formula?

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

When do you use Binomial Coefficient?

Use Pascal's triangle for small values: each entry is the sum of the two entries above it. For calculation, cancel common factors before multiplying to keep numbers manageable.

Prerequisites

combinationfactorial

How Binomial Coefficient Connects to Other Ideas

To understand binomial coefficient, you should first be comfortable with combination and factorial. Once you have a solid grasp of binomial coefficient, you can move on to binomial theorem and binomial distribution.

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