Multinomial Coefficient Calculator

How the multinomial coefficient calculator works

A multinomial coefficient counts distinct arrangements when a total set is divided among several category sizes or contains repeated item types. It generalizes the binomial coefficient: take factorial of the total count and divide by factorial of every category count. This calculator displays numerator and denominator product, making repeated-category corrections visible instead of presenting only the final count.

The default category counts 2, 1, 1 total four items. There are 4 factorial raw orders, but swapping the two items in the repeated category does not make a new arrangement, so divide by 2 factorial. The verified coefficient is 24 divided by 2, or 12 distinct arrangements.

Reading multinomial coefficient calculator results privately

Counts must be nonnegative integers and their total is capped at 18 so factorial arithmetic remains an exact safe integer. Category labels do not matter; only their sizes affect the coefficient. The result counts allocations or arrangements under the multinomial model and does not attach probabilities until category probabilities are supplied separately. This multinomial coefficient calculator calculation runs entirely in your browser, so the numbers you enter never leave your device.