Quick answer

Compute C(n, k) by multiplying k descending factors from n, dividing by k!, or by using COMBIN/nCr tools after you confirm k ≤ n.

Formula

  • Hand: C(n,k) = (n)…(n−k+1) / k!
  • Sheets: =COMBIN(n,k)

Introduction

Start every problem by naming n and k in plain language. If you cannot do that, pause and reread the prompt before any arithmetic.

The formula explanation article is the reference when you forget whether to divide by k! or (n − k)!.

When numbers grow, switch from paper to the calculator guide on this site so you still get exact integers without floating-point rounding.

What you are calculating

Each method below answers the same question: how many unordered selections of size k from n labeled objects? Methods differ in speed and error profile, not in meaning.

Independent methods should agree. If hand work disagrees with COMBIN, assume the hand work is wrong until you find the slip.

Method comparison

  • Manual product + division: best for learning
  • Calculator nCr / COMB: fast for exams
  • Spreadsheet COMBIN: audit tables
  • Pascal recursion: small n pattern practice

Manual calculation builds structure. Write the k-factor numerator explicitly, write k!, cancel, then multiply. Skipping the write-up step is how students lose factors of 2.

Scientific calculators expose nCr as a two-argument function. Enter n, then k, read the integer. Graphing models may display scientific notation for large results even when the value is exact.

Step-by-step guide

  1. Manual: build and cancel. Example C(8,3): numerator 8×7×6, denominator 6, result 56.
  2. Calculator: use nCr with integer inputs. Re-enter if you see a decimal with a long tail; large exact answers should still be integers.
  3. Spreadsheet: COMBIN(n,k). Drag formulas down columns when you compare many pairs for lab reports.
  4. Shortcut: symmetry and Pascal. Use C(n,k)=C(n,n−k) when k > n/2. Use triangle rows when n ≤ 12 and you want visual checks.

Classroom drill: C(14, 5)

Symmetry: C(14,5) = C(14,9). Computing C(14,5) is smaller. Numerator five factors: 14×13×12×11×10 = 240240. Divide by 120 to get 2002.

Calculator check: nCr(14,5) should read 2002 exactly. Spreadsheet =COMBIN(14,5) matches.

If all three disagree, trace cancellation on paper before blaming the device.