is this correct? $\binom{n}{k}=\binom{n}{k-1}-\binom{n-1}{k-1}$?

Question

My instructor wrote

$$\binom{n}{k}=\binom{n}{k-1}-\binom{n-1}{k-1}$$

I doubt it is incorrect.

Is it?

Use the binomial coefficient definition of "n choose k" (being n factorial divided by k factorial times n minus k factorial) and you can use some algebra and try to establish the equivalence. — Jon Staggs, Jun 20 '17 at 16:30
It might be good to say why you doubt it's correct (unless you're skeptical of everything your professor writes) -- did you see a different formula elsewhere? Did you try an example that didn't work? Edit: although now I see you "doubt it is incorrect", so the whole thing is odd. — pjs36, Jun 20 '17 at 16:33
I am wondering if your prof meant to write something like. $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ — Doug M, Jun 20 '17 at 16:58

score 3 · Answer 1 · answered Jun 20 '17 at 16:48

3

$$3=\binom{3}{2}\neq\binom31-\binom21=3-2=1$$

answered Jun 20 '17 at 16:48

windircurse

score 1 · Answer 2 · answered Jun 20 '17 at 16:38

1

Search Pascal's rule , i think your assumption is wrong. https://en.wikipedia.org/wiki/Pascal%27s_rule

answered Jun 20 '17 at 16:38

WhiteHole

Here's a MathJax tutorial :) – Shaun Jun 20 '17 at 16:42
1

thank you so much , i didn't know anything , i couldn't even write the answer. – WhiteHole Jun 21 '17 at 15:39

2 Answers2