Five choose six equals zero

Algebra Level 4

Let $h(x)$ denote a monic $8^\text{th}$ -degree polynomial such that $h(m) = \binom m6$ for $m = 6,7,8,\ldots,13$ .

Find $h(5)$ .

Notation : $\binom mn$ denotes the binomial coefficient $\binom mn = \frac{m!}{n!(m-n)!}$ for non-negative integers $m\geq n$ .

The answer is 40320.

1 solution

Rishabh Jain
Feb 27, 2016

$h(x)=(x-6)(x-7)(x-8)\cdots(x-13)+\binom{x}{6}$ $\implies h(5)=8!+\binom{5}{6}=8!=\Large \boxed{\color{#20A900}{40320}}$ Note:- $\binom{5}{6}$ can be thought of number of ways of selecting 6 out of 5 objects which are $\boxed 0$ .

Great thanks! By the way, we're going to do a wiki collaboration for the wiki page . Are you interested in joining?

Pi Han Goh - 5 years, 3 months ago

Unfortunately I'm having scarcity of time these days so I'll say No .... but surely in future I'll contribute for brilliant whenever I'll get time :-)

Rishabh Jain - 5 years, 3 months ago

Oh sure sure. By the way, if you change your mind. we will be at here to discuss the wiki collab. Which will take place on 9.30pm IST today.

Pi Han Goh - 5 years, 3 months ago

Wait... isn't $h(t)$ a polynomial of degree 8?

Otto Bretscher - 5 years, 3 months ago

Awwwhhhhh... Let me fix that.

Pi Han Goh - 5 years, 3 months ago

It's one of those funny situations where the author and the "solver" of the problem make the same mistake ;)

Otto Bretscher - 5 years, 3 months ago

@Otto Bretscher – I can't count above five. Sorry.

Pi Han Goh - 5 years, 3 months ago

@Pi Han Goh – I had to use my fingers myself ;) Luckily, you didn't go from 6 to 2016

Otto Bretscher - 5 years, 3 months ago

@Otto Bretscher – Way too difficult for me to do by hand. WolframAlpha is needed.

Pi Han Goh - 5 years, 3 months ago