We have n objects, labeled with k different labels; this includes n1 of the first type, n2 of the second, …, nk of the kth type. The number of ways that our objects can be labeled equals n!/(n1!∙n2!∙...∙nk!). We call this expression the multinomial formula. Note that n1 + n2 + … + nk = n.
Consider a box with two red balls, three green balls, and one white ball. The red balls are marked with the numbers 1 and 2; the green balls with the numbers 3, 4, and 5; and the white ball is marked with the number 6. In how many different ways can the balls be arranged?
There are n1 = 2 red balls, n2 = 3 green balls, and n3 = 1 white ball. Altogether we have n = n1 + n2 + n3 = 2 + 3 + 1 = 6 balls. They can be arranged in
n!/(n1!∙n2!∙...∙nk!) = 6!/(2!∙3!∙1!)