Binomial coefficient formula: Why the dividend is a multiple of divisor

Asked Jul 13 '13 at 05:22

Active Jul 13 '13 at 05:46

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Looking at the formula for binomial co-effient,

$$ \binom{n}{r}= \frac {n(n-1)...(n-r+1)}{ 1(2)(3)...(r)} $$

I am wondering why $ n(n-1)...(n-r+1) $ is a multiple of $ 1(2)(3)...(r) $ .

I understand from the applications of this formula that only integer values makes sense as in $ n C r $. But, why are these two products be related in such a way, purely thinking of them as product of numbers and from algebra, how can we prove that they will always be related that way.

asked Jul 13 '13 at 05:22

Suresh

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http://math.stackexchange.com/questions/11601/proof-that-a-combination-is-an-integer – lab bhattacharjee Jul 13 '13 at 05:24
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It is true because ${n \choose r}$ has an independent definition that makes it an integer (namely it is the number of subsets of an $n$-element set of size $r$). – Qiaochu Yuan Jul 13 '13 at 05:24

Binomial coefficient formula: Why the dividend is a multiple of divisor

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