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Is it appropriate to describe the function $$f(x) = \binom{x}{2}$$ as parabolic? How does it vary over $c$: $$f(x) = \binom{x}{c}?$$ And, is there any intuition behind these graphs? For example, their resemblance to the polynomials of degree $c?$

I tried inputting one into wolfram alpha and it spat out a continuous function, which confused me. This is what prompted me to ask the question.

Rafael Vergnaud
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3 Answers3

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Yes, you can generalise the factorial, choose function, etc. to not whole numbers too. $$\binom{x}{k}=\frac{x!}{k!(x-k)!}$$ $$=\frac{x(x-1)\dots(x-k)!}{k!(x-k)!}$$ $$=\frac{x(x-1)\dots(x-k+1)}{k!}$$ So for $k=2$ you will get: $$\binom x2=\frac{x(x-1)}{2!}=\frac{x^2-x}{2}$$ And for $k=3$: $$\binom x3=\frac{x(x-1)(x-2)}{3!}=\frac{x^3-3x^2+2x}{6}$$ And so on.

Joffan
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Botond
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    To add: the polynomial that shows up in the numerator of these expressions is what's often called a falling factorial, $x^{(k)}$. Thus, we have the relationship $\dbinom{x}{k}=\dfrac{x^{(k)}}{k!}$. – J. M. ain't a mathematician Mar 17 '18 at 01:12
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The function $B_c:\mathbb N\to\mathbb N$ defined by $$n\mapsto {n\choose c}$$ is a polynomial of degree $\min(c,n-c)$. So it is "parabolic" (assuming you mean quadratic) when $c=2$ or $c=n-2$.

And be careful when you define functions, it is of good habits to specify the sets where it is defined as well as the image set.

If you want a function defined for real $x$, search for an analytic continuation of the binomial.

Tal-Botvinnik
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I have usually used the binomial formula with discrete whole numbers. If you want to plot this, you get points on a grid.

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Mark R
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  • So, it is not appropriate to enter in real number into x? I tried inputting it into wolfram alpha and it spat out continuous functions, which confused me. This is what prompted me to ask the question. – Rafael Vergnaud Mar 16 '18 at 22:13
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    It can be extended to real numbers with the gamma function, and in this case it has an obvious extension to polynomials. – Acccumulation Mar 16 '18 at 22:17
  • The binomial formula can be written as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. Taking factorials require the input to be an positive integer value. – Mark R Mar 16 '18 at 22:19
  • @MarkR you can generalise the factorial for complex numbers too... See: https://en.wikipedia.org/wiki/Gamma_function – Botond Mar 16 '18 at 22:21
  • @MarkR For example, $i!$: http://www.wolframalpha.com/input/?i=factorial(i) – Botond Mar 16 '18 at 22:22
  • Neat, I have not seen it being used in that way before. Learn something new every day. – Mark R Mar 16 '18 at 22:24
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    if the binomial is defined through the falling factorial (as in Botond answer) then you do not need to invoke gamma, and $\binom{x}{q}$ is a polynomial of degree $q$, for $0 \le q \in \mathbb Z$ – G Cab Mar 16 '18 at 22:24
  • "The binomial formula is not a continuous function." False! if all the points of the domain are isolated, the function is trivially continuous, – Martín-Blas Pérez Pinilla Mar 16 '18 at 22:32