Condition for Equality of Adjacent Binomial Coefficients
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Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$.
Then:
- $\dbinom n k = \dbinom n {k + 1}$
- $n$ is an odd integer
- $k = \dfrac {n - 1} 2$
Proof
Sufficient Condition
Let $n$ be odd and $k = \dfrac {n - 1} 2$.
Let $n = 2 m + 1$ for some $m \in \Z_{\ge 0}$.
We have:
\(\ds k\) | \(=\) | \(\ds \dfrac {n - 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2 m + 1} - 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 m} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m\) |
Hence:
\(\ds \dbinom n k\) | \(=\) | \(\ds \dbinom {2 m + 1} m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {2 m + 1} {\paren {2 m + 1} - m}\) | Symmetry Rule for Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {2 m + 1} {m + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom n {k + 1}\) |
$\Box$
Necessary Condition
Let $n$ and $k$ be such that:
- $\dbinom n k = \dbinom n {k + 1}$
From Condition for Increasing Binomial Coefficients, it is not the case that:
- $0 \le k < \dfrac {n - 1} 2$
as under such a condition we would have:
- $\dbinom n k = \dbinom n {k + 1}$
So:
- $k \ge \dfrac {n - 1} 2$
By the Symmetry Rule for Binomial Coefficients, we also have that:
- $\dfrac {n - 1} 2 < k \le n \iff \dbinom n k > \dbinom n {k + 1}$
and so for $\dbinom n k = \dbinom n {k + 1}$ it is not the case that $\dfrac {n - 1} 2 < k \le n$.
So:
- $k \le \dfrac {n - 1} 2$
Hence that means:
- $k = \dfrac {n - 1} 2$
which means:
- $n = 2 k + 1$
and so $n$ is odd.
$\blacksquare$
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem: Problems $1.2$: $1 \ \text{(b)}$