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TL;DR

The number \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), representing the number of ways to choose k items from n. Generalized to real-valued n via the Gamma function.

By Valenke Exam Prep Team·Last updated 2026-06-03

Binomial Coefficient

The number \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), representing the number of ways to choose k items from n. Generalized to real-valued n via the Gamma function.

Why it matters for interviews

Ubiquitous in probability (binomial distribution), combinatorics (counting subsets), and finance (binomial tree pricing). Fluency with binomial coefficient identities is essential for quant interviews.

Definition and Mathematical Foundation

The number \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), representing the number of ways to choose k items from n. Generalized to real-valued n via the Gamma function.

Application in Quantitative Finance

Ubiquitous in probability (binomial distribution), combinatorics (counting subsets), and finance (binomial tree pricing). Fluency with binomial coefficient identities is essential for quant interviews.

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Frequently Asked Questions

What are the most important binomial coefficient identities?▼
Pascal's identity, Vandermonde's identity, the hockey stick identity \( \sum_{i=r}^n \binom{i}{r} = \binom{n+1}{r+1} \), and the symmetry \( \binom{n}{k} = \binom{n}{n-k} \).
How does the binomial coefficient appear in the binomial distribution?▼
\( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \). The coefficient counts the number of ways to arrange k successes in n trials.
What is the connection to the binomial tree in finance?▼
In the Cox-Ross-Rubinstein model, the price after n steps has \( \binom{n}{k} \) paths reaching k up-moves. Option prices are computed as binomial-weighted sums of terminal payoffs.