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202110110958 Sidenote (Section 3.8.2)

Given that the binomial coefficient is so defined as

\begin{pmatrix} N \\ n \end{pmatrix} \stackrel{\textrm{def}}{=} \displaystyle{\frac{N!}{n!(N-n)!}}.

Prove the identities (3.49), (3.50), and (3.51) in Section 3.8.2 Useful Identities for the Binomial Coefficients. Namely,

Eq. (3.49):

\begin{pmatrix} N \\ 0 \end{pmatrix} = \begin{pmatrix} N \\ N \end{pmatrix} = 1

Eq. (3.50):

\begin{pmatrix} N - 1 \\ n \end{pmatrix} + \begin{pmatrix} N - 1 \\ n-1 \end{pmatrix} = \begin{pmatrix} N \\ n \end{pmatrix}

Eq. (3.51):

\begin{pmatrix} N \\ n+1 \end{pmatrix} = \displaystyle{\frac{N-n}{n+1}}\begin{pmatrix} N \\ n \end{pmatrix}

R. H. Swendsen. (2012). An Introduction to Statistical Mechanics and Thermodynamics

Proof of Eq. (3.49).

\begin{aligned} \begin{pmatrix} N \\ 0 \end{pmatrix} & = \frac{N!}{0!(N-0)!} \\ & = \frac{N!}{1\times N!} \\ & = 1 \end{aligned}
\begin{aligned} \begin{pmatrix} N \\ N \end{pmatrix} & = \frac{N!}{N!(N-N)!} \\ & = \frac{N!}{N!\times 1} \\ & = 1 \end{aligned}

Proof of Eq. (3.50).

\begin{aligned} & \quad \begin{pmatrix} N-1 \\ n \end{pmatrix} + \begin{pmatrix} N-1 \\ n-1 \end{pmatrix} \\ & = \frac{(N-1)!}{(n)!(N-1-n)!} + \frac{(N-1)!}{(n-1)!\big((N-1)-(n-1)\big)!} \\ & = \frac{(N-1)!}{(n)!(N-1-n)!} + \frac{(N-1)!}{(n-1)!(N-n)!} \\ & = \frac{(N-n)(N-1)!+(n)(N-1)!}{(n)(n-1)!(N-n)(N-n-1)!} \\ & = \frac{N(N-1)!}{n!(N-n)!} \\ & = \frac{N!}{n!(N-n)!} \\ & = \begin{pmatrix} N \\ n \end{pmatrix} \\ \end{aligned}

Proof of Eq. (3.51).

\begin{aligned} &\quad \begin{pmatrix} N \\ n+1 \end{pmatrix} \\ & = \frac{N!}{(n+1)!\big(N-(n+1)\big) !} \\ & = \frac{(N!)(N-n)}{\big( (n+1)(n!)\big) \big( (N-n)(N-n-1)! \big)} \\ & = \bigg(\frac{N-n}{n+1}\bigg)\bigg(\frac{N!}{n!(N-n)!}\bigg) \\ & = \frac{N-n}{n+1}\begin{pmatrix} N \\n \end{pmatrix} \\ \end{aligned}

QED

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