10 Choose 5 = 252
There are 252 ways to choose 5 items from 10 when order does not matter. It is the largest binomial coefficient for n = 10, the peak of the tenth row of Pascal’s triangle.
Combinations (10C5)
252
Permutations (10P5)
30,240
How 10 choose 5 is calculated
The combinations formula is C(n, r) = n! / (r! × (n − r)!). Substituting n = 10 and r = 5 gives:
C(10, 5) = 10! / (5! × 5!)
= 30,240 / 120
= 252The 5! in the denominator cancels all but the top 5 factors of 10!, so 10 choose 5 is the product of 5 descending terms starting at 10, divided by 5!. That product, 30,240, is the number of ordered arrangements (10P5); dividing by 5! = 120 removes the duplicate orderings and leaves 252.
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