# Thread: binomial and multinomial theorms

1. ## binomial and multinomial theorms

Suppose that n1,n2,....., nk are non-negative integers such that
n1 +n2 +....nk = n.
(a) Use induction to show that, for every integer k>= 1,

(nCn1)(n-n1Cn2)(n-n1-n2Cn3).....(n-n-1-n2-......-nk-1Cnk) = n!/(n1!n2!...nk!)
[IMG]file:///C:/temp/moz-screenshot.jpg[/IMG][IMG]file:///C:/temp/moz-screenshot-1.jpg[/IMG](view another form of this equation by viewing the attachment)

(b) Suppose you have n1 identical balls of colour 1, n2 identical balls of
colour 2, and so on until, finally, you have nk identical balls of colour
k. Count the number of arrangements of these balls in a line in two
different ways and, by doing so, obtain a combinatorial proof of the
identity in part (a).