Short answer
Same number, different framing
Combination describes the counting idea: choose r items from n when order does not matter and repetition is not allowed.
Binomial coefficient describes the same number in notation form: C(n, r) or nCr.
Equivalent forms
These mean the same thing
8C4C(8, 4)- 8 choose 4
All three ask for the same value. They count unordered groups of 4 taken from 8 distinct items, so each expression evaluates to the same answer.
Why both names exist
Counting language vs algebra language
The word combination usually appears in counting and word-problem contexts. The phrase binomial coefficient usually appears in algebra, probability, Pascal’s triangle, and the binomial theorem.
That is why sites often keep both pages or calculators even though the math is the same. The user intent is different even when the engine underneath is identical.
When to use which page
Choose the page that matches the search language
- Use the Combination Calculator when the question is phrased as choosing items or making groups.
- Use the Binomial Coefficient Calculator when the query is written as
nCr,C(n, r), or a phrase like “8c4 calculator.”
Formula
The notation and the combination formula match
C(n, r) = nCr = n! / (r!(n - r)!)
So if you already know a plain combination formula, you already know the binomial coefficient formula too.