Stuck with a proof, please help :) (1 Viewer)

Elise8842 · Jan 10, 2015

Hi,

I'm currently stuck with the proof question in the first picture, my working out is shown in the second picture and the book gives a hint(in the 3rd pic)

It would be lovely if someone could show me how to continue the proof XD

Thanks in advance!

~

Trebla · Jan 10, 2015

Your working is fine, it's just a matter of simplifying. Note that the last factor of product is 1, which is clear from the hint or intuitively from the fact that there is only one combination of the remaining objects in the last pile.

$\binom{n}{r_1}\times\binom{n-r_1}{r_2}\times....\times\binom{n-r_1-r_2-.....-r_{k-2}}{r_{k-1}}\times\binom{n-r_1-r_2-....-r_{k-1}}{r_k}\\\\=\binom{n}{r_1}\times\binom{n-r_1}{r_2}\times.....\times\binom{n-r_1-r_2-....-r_{k-2}}{r_{k-1}}\\\\=\dfrac{n!}{r_1!(n-r_1)!}\times\dfrac{(n-r_1)!}{(n-r_1-r_2)!r_2!}\times\dfrac{(n-r_1-r_2)!}{(n-r_1-r_2-r_3)!r_3!}\times....\times\dfrac{(n-r_1-r_2-....-r_{k-2})!}{(n-r_1-r_2-....-r_{k-2}-r_{k-1})!r_{k-1}!}\\\\=\dfrac{n!}{r_1!r_2!...r_{k-1}!(n-r_1-r_2-....-r_{k-2}-r_{k-1})!}\\\\=\dfrac{n!}{r_1!r_2!...r_k!}$

Stuck with a proof, please help :) (1 Viewer)

Elise8842

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Trebla

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