Quick answer
n choose r means C(n, r) = n! / (r! (n − r)!).
Formula
- P(n,r) = n!/(n−r)! counts ordered picks
- C(n,r) = P(n,r)/r!
Introduction
On this site the live tool labels the fields n and k. Treat k as r whenever your textbook uses r for successes or selections.
If order matters, stop reading about choose r and open permutations vs combinations for side-by-side formulas and a decision checklist.
After notation clicks, practice arithmetic in how to calculate binomial coefficients so calculator notation and hand work stay aligned.
Symbols on calculators and papers
TI-style devices use nCr. Spreadsheet functions use COMBIN(n, r). Textbooks write (n r) with a vertical stack. All request the same integer when inputs are valid.
Phrase "choose" signals combinations. Phrase "arrange" or "rank" signals permutations. Mixed phrases appear in contest problems to test reading care.
Connecting permutations and combinations
- P(n, r) = n × (n − 1) × … × (n − r + 1)
- C(n, r) = P(n, r) / r!
Every ordered r-sequence collapses r! ways into one combination. That division is the bridge formula students forget when they memorize pieces separately.
Step-by-step guide
- Read n as pool size. Total distinct items available.
- Read r as selection size. How many you pick without caring about order inside the pick.
- Apply C(n, r) or nCr. Confirm r ≤ n before you press enter.
Same story, two counts
From 5 servers, pick 3 for a maintenance window where order does not matter: C(5, 3) = 10.
If the same 3 servers must be visited in a mandated sequence, ordered assignments use P(5, 3) = 60.
The ratio 60/10 = 6 = 3! illustrates the r! factor between models.
