Wednesday, October 9, 2019

combinatorics - Prove that 00=1 using binomial theorem



I have read in Donald Knuth's book that 00=1. And it has been said to come from the basic formula of (x+y)r. Can anyone prove how it comes?


Answer




Consider
(x+0)^n=\color{#00A000}{\binom{n}{0}x^n0^0}+\color{#C00000}{\binom{n}{1}x^{n-1}0^1+\dots+\binom{n}{n}x^00^n}
Since all the red terms are 0, for the left side, x^n, to equal the right side, x^n0^0, we need 0^0=1.



Furthermore, we have
\sum_{k=0}^n(-1)^k\binom{n}{k}=(1-1)^n

Note that this holds for n=0 only if 0^0=1.


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