For my Math Physics homework, I have to show that ${-1\choose n} = (-1)^n$. I read the textbook (no help) and this wasn't covered in lecture, I have no idea where to even begin. Please help.
2026-04-19 21:07:41.1776632861
Prove ${-1 \choose n} = (-1)^n$
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With positive integers, one can say things like $$ \binom 8 3 = \frac{8!}{3!5!}. $$ By canceling $5!$ one can say $$ \binom 8 3 = \frac{8\cdot7\cdot6}{3\cdot2\cdot1}. $$ This latter form works if in place of $8$ we put a number that is not a nonnegative integer, even if we have not defined factorials for those: $$ \binom{8.1}3 = \frac{(8.1)(7.1)(6.1)}{3\cdot2\cdot1}. $$ This can be applied to negative integers: $$ \binom{-8}{3} = \frac{(-8)(-9)(-10)}{3\cdot2\cdot1}. $$ So we have $$ \binom{-1}3 = \frac{(-1)(-2)(-3)}{3\cdot2\cdot1}. $$ Now think about what happens if $n=3$ is changed to $n=\text{some other positive integer}$.
The number $$ \left|\binom{-8}{3}\right| = \left|\frac{(-8)(-9)(-10)}{3\cdot2\cdot1}\right| $$ is how many sub-multisets of size $3$ are in a set of size $8$. I've seen it expressed loosely as "negative sets are multisets".