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Hello. I was looking for the proof of binomial theorem by mathematical induction and I reached this step . I just don't understand why the terms underlined red are canceled out .

  • check the definition of a binomial coefficient $n \choose k$ when $k < 0$ or $n < k$ and you will see – fonfonx Jul 24 '17 at 03:00
  • Can I enter the minus inside the binomial coefficient ? –  Jul 24 '17 at 03:05
  • Okay thank you now I understood the reason they canceled out , but still need to know if I could enter the minus inside it –  Jul 24 '17 at 03:08
  • what do you mean by "enter the minus inside it"? – fonfonx Jul 24 '17 at 03:09
  • $$\binom{a}{b} = \begin{cases} \frac{a!}{b!(a-b)!} & \textrm{if } b\geq0,;a\geq b\ 0 & \textrm{otherwise} \end{cases}$$ – ChargeShivers Jul 24 '17 at 03:11
  • Thank you . I mean by entering the minus that the terms in the picture have a minus before them . Can I put the minus outside , inside the binomial coefficient? –  Jul 24 '17 at 07:05

1 Answers1

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We define for $r\in\mathbb{R}$ and $k\in\mathbb{Z}$ \begin{align*} \binom{r}{k}= \begin{cases} \frac{r(r-1)\cdots (r-k+1)}{k(k-1)\cdots 3\cdot2\cdot1}&\qquad k\geq 0\tag{1}\\ 0&\qquad k<0 \end{cases} \end{align*}

We obtain from (1) \begin{align*} \binom{n-1}{n}=\frac{(n-1)(n-2)\cdots 2\cdot 1\cdot \color{blue}{0}}{n(n-1)\cdots 3\cdot 2\cdot 1}=0 \end{align*}

and \begin{align*} \binom{n-1}{-1}=0 \end{align*}

since the lower entry $-1$ is negative.

Hint: You might find chapter $5$ Binomial Coefficients (definition above is (5.1)) in Concrete Mathematics by R.L.Graham, D.E. Knuth and O. Patashnik helpful.

Markus Scheuer
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