Binomial Coefficient with One

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Theorem

$\forall r \in \R: \dbinom r 1 = r$

where $\dbinom r 1$ denotes a binomial coefficient.


The usual presentation of this result is:

$\forall n \in \N: \dbinom n 1 = n$


Proof

From the definition of binomial coefficients:

$\dbinom r k = \dfrac {r^{\underline k}} {k!}$ for $k \ge 0$

where $r^{\underline k}$ is the falling factorial.

In turn:

$\displaystyle x^{\underline k} := \prod_{j \mathop = 0}^{k - 1} \left({x - j}\right)$

But when $k = 1$, we have:

$\displaystyle \prod_{j \mathop = 0}^0 \left({x - j}\right) = \left({x - 0}\right) = x$

So:

$\forall r \in \R: \dbinom r 1 = r$

$\blacksquare$


This is completely compatible with the result for natural numbers:

$\forall n \in \N: \dbinom n 1 = n$

as from the definition:

$\dbinom n 1 = \dfrac {n!} {1! \ \left({n - 1}\right)!}$

the result following directly, again from the definition of the factorial where $1! = 1$.

$\blacksquare$


Also see


Sources