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50 Choose 5 = 2,118,760

There are 2,118,760 ways to choose 5 items from 50 when order does not matter. It is the number of main-number combinations in a 5-from-50 draw such as EuroMillions, before the extra balls.

Combinations (50C5)
2,118,760
Permutations (50P5)
254,251,200

How 50 choose 5 is calculated

The combinations formula is C(n, r) = n! / (r! × (n − r)!). Substituting n = 50 and r = 5 gives:

C(50, 5) = 50! / (5! × 45!)
         = 254,251,200 / 120
         = 2,118,760

The 45! in the denominator cancels all but the top 5 factors of 50!, so 50 choose 5 is the product of 5 descending terms starting at 50, divided by 5!. That product, 254,251,200, is the number of ordered arrangements (50P5); dividing by 5! = 120 removes the duplicate orderings and leaves 2,118,760.

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