Quick answer
C(n, k) is the number of k-element subsets of an n-element set. Equivalently, it is "n choose k," written (n k) or nCk on calculators.
Formula
- C(n, k) = n! / (k! (n − k)!)
- Valid for integers with 0 ≤ k ≤ n.
Introduction
Most students first meet binomial coefficients as mysterious parentheses in the binomial theorem, then see the same numbers again in probability homework. The underlying idea is simpler than the notation suggests: you are counting subsets, not arranging trophies on a podium.
Once you can state n and k in words, the binomial coefficient formula article shows how factorials and symmetry turn that story into a number you can check on paper.
Before you memorize anything, confirm the problem ignores order. If rearranging the same picks creates a new outcome, you are in permutation territory, and our guide on permutations vs combinations walks through when to switch formulas.
Definition and meaning
Imagine five distinct books on a shelf and you plan to slip two into your bag. How many different bags are possible if you only care which titles you took, not the order you grabbed them? Listing pairs builds intuition: there are ten pairs, and that count is C(5, 2).
Formally, n is the size of the labeled pool and k is how many labels you select. Two selections are identical when they contain exactly the same elements. ABC and CBA describe the same committee even though they are different permutations.
The phrase "binomial" reflects the link to expanding (a + b)^n, where each term's coefficient counts how many sign choices produce a given power. You do not need algebra to use C(n, k) in pure counting problems, but the connection explains why the same numbers appear in discrete math, statistics, and computer science curricula.
Combination notation you will see
- C(n, k) common function notation
- (n k) textbook combination symbol
- nCk calculator shorthand
- n choose k spoken form
All of these refer to the same integer when k is between 0 and n. Writers swap k for r in probability texts ("n choose r"), which confuses beginners until they realize r is just the selected count.
Edge cases are worth memorizing because they appear on every exam: C(n, 0) = 1 (empty selection), C(n, n) = 1 (take everything), and C(n, 1) = n (pick one labeled item). When k > n, the count is zero because you cannot choose more distinct items than exist.
Step-by-step guide
- Translate the story into n and k. n counts distinct objects in the full set. k counts how many you select when order is irrelevant. If the wording mentions committees, samples without replacement labels, or choosing winners, you are usually on the right track.
- Verify that order truly does not matter. Ask whether swapping two chosen items changes the outcome. If yes, stop and use permutations instead of C(n, k).
- Check the constraints 0 ≤ k ≤ n. Negative k or k larger than n are not valid in standard combinatorics problems unless the question explicitly extends definitions.
- Compute or verify with a trusted tool. Use the product formula by hand for small n, then confirm with the home-page calculator so arithmetic slips do not stick.
Worked example: choosing two projects
A manager must pick two of six distinct proposals for a pilot program. Order of selection does not matter, so n = 6 and k = 2.
By listing, or by formula, C(6, 2) = 15 possible pairs. If the manager later ranks the two pilots, that ranking step is separate and would use permutations on the chosen pair.
Real-world applications repeat the same pattern: lottery lines (large n, small k), review panels, and sampling frames in survey design all reduce to counting subsets before any probability model attaches weights.
