Upper Bound for Binomial Coefficient
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Theorem
Let $n, k \in \Z$ such that $n \ge k \ge 0$.
Then:
- $\dbinom n k \le \left({\dfrac {n e} k}\right)^k$
where $\dbinom n k$ denotes a binomial coefficient.
Proof
From Lower and Upper Bound of Factorial, we have that:
- $\dfrac {k^k} {e^{k - 1} } \le k!$
so that:
- $(1): \quad \dfrac 1 {k!} \le \dfrac {e^{k - 1} } {k^k}$
Then:
\(\ds \dbinom n k\) | \(=\) | \(\ds \dfrac {n^\underline k} {k!}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac {n^k} {k!}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac {n^k e^{k - 1} } {k^k}\) | from $(1)$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac {n^k e^k} {k^k}\) |
Hence the result.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $67$