Replacement Calc

When picking n items out of N total items, where m of them are distinct, the odds of picking exactly k distinct items is defined as:
P(X = k) = _mC_k * _N-mC_n-k / _NC_n

Where _nC_x ("n choose x") is defined as
_nC_x = n! / [ x! (n - x)! ]

Mean: n * m / N

Inputs should be positive integers. Non-integer inputs will be rounded down.

Picking wihtout replacement means that once you've picked your item, you do not put it back into the pool of items to pick before picking the next item

Wikipedia on Hypergeometric Distributions

Wikipedia on Simple Random Samples

Wikipedia on the Urn Problem

UTexas provides formulas for sampling with and without replacement

• exactly k items is represented in blue,
• less than k items is represented in red, and
• greater than k items is represented in green

Note that we limit input to 1000. This is because when calculating n choose k for inputs of 1000 and 500, we get about 1e300, which is a number larger than should ever been needed, and is just approaching the largest number Javascript will store.

To put this into perspective: Say we take every particle in the known universe and multiply that number by the number of nanoseconds the universe has been alive. This number squared is significantly less than 1e300.