DISCRETE MATH: Binomial Theorem proof (using Corollary 2)

VinnyCee

1. The problem statement, all variables and given/known data

Show that if n is a positive integer, then [tex]1\,=\,\binom{n}{0}\,<\,\binom{n}{1}\,<\,\cdots\,<\,\binom{n}{\lfloor\fr ac{n}{2}\rfloor}\,=\,\binom{n}{\lceil\frac{n}{2}\rceil}\,>\,\cdots\,>\b inom{n}{n\,-\,1}\,>\,\,\binom{n}{n}\,=\,1[/tex]



2. Relevant equations

I think this proof involves corollary 2 of the Binomial Theorem.

[tex]\sum_{k\,=\,0}^n\,(-1)^k\,\binom{n}{k}\,=\,0[/tex]



3. The attempt at a solution

I have NO idea where to start a proof of this sort. This problem is labeled as 'easy', but I don't see the easiness yet. I know that an attempt at a solution is required before any assistance is rendered, but I have been staring at (and thinking about) this problem for two days now. I can't think of ANYTHING that is even remotely helping me to "show" this theorem.

I am not trying to have someone else do my homework or anything like that, I just need a pointer or two. If you don't believe me, look at my previous threads, all have detailed attempts and most were completed problems after help was given.

Thanks for your help in advance.

mjsd

have you tried using the definition
[tex]\binom{n}{k}=\frac{n!}{k!(n-k)!}[/tex]
to help?

VinnyCee

Yeah, I used that up until the ceiling and floor functions in the middle. I can show that 1 is equal to the next term and that the term after that is indeed larger than the term equal to one. But what do I do with the ceiling and floor functions that come after? How to show that they are larger?

mjsd

what can you and what can't you prove? you can't prove for the cases k > n/2 or k < n/2 or just the middle terms ?

VinnyCee

By definition, [itex]\binom{n}{0}\,=\,1[/itex]. Now, [itex]\binom{n}{1}\,=\,\frac{n!}{(n\,-\,1)!}[/itex]. And by definition of the factorial function [itex]n!\,>\,(n\,-\,1)![/itex]. Also, a divisor that is less than the dividend results in a quotient that is greater than one. This proves up to the point of [itex]\binom{n}{1}[/itex], and I think it can be extended up to the part where I am stuck. How do I work with the floor functions using the definition of binomial coefficient?

[tex]\binom{n}{\lfloor\frac{n}{2}\rfloor}[/tex]

VinnyCee

I guess I have to consider two cases of the center equality (the one using the ceiling and floor functions) in order to prove the whole inequality. There is a case of when [itex]n[/itex] is even and the case of when [itex]n[/itex] is odd. In the even case, the two terms are obviously equal.


[itex]n[/itex] = 4:

[tex]\binom{4}{\lfloor\frac{4}{2}\rfloor}\,=\,\binom{4}{2}\,=\,\frac{4!}{2!( 4\,-\,2)!}\,=\,\frac{4!}{2!}\,=\,\frac{24}{2}\,=\,12[/tex]

[tex]\binom{4}{\lceil\frac{4}{2}\rceil}\,=\,\binom{4}{2}\,=\,\frac{4!}{2!(4\ ,-\,2)!}\,=\,\frac{4!}{2!}\,=\,\frac{24}{2}\,=\,12[/tex]


Okay, but how do I prove anything about the odd [itex]n[/itex]-values?

[itex]n[/itex] = 5:

[tex]\binom{5}{\lfloor\frac{5}{2}\rfloor}\,=\,\binom{5}{2}\,=\,\frac{5!}{2!( 5\,-\,2)!}\,=\,\frac{5!}{2!\,3!}\,=\,\frac{120}{12}\,=\,10[/tex]

[tex]\binom{5}{\lceil\frac{5}{2}\rceil}\,=\,\binom{5}{3}\,=\,\frac{5!}{3!(5\ ,-\,3)!}\,=\,\frac{5!}{2!\,3!}\,=\,\frac{120}{12}\,=\,10[/tex]


[itex]n[/itex] = 7:

[tex]\binom{7}{\lfloor\frac{7}{2}\rfloor}\,=\,\binom{7}{3}\,=\,\frac{7!}{3!( 7\,-\,3)!}\,=\,\frac{7!}{3!\,4!}\,=\,\frac{5040}{144}\,=\,35[/tex]

[tex]\binom{7}{\lceil\frac{7}{2}\rceil}\,=\,\binom{4}{4}\,=\,\frac{7!}{4!(7\ ,-\,4)!}\,=\,\frac{7!}{4!\,3!}\,=\,\frac{5040}{144}\,=\,35[/tex]


I guess that proves that in either case the two are equal! Now how do I combine it all into a concise proof?

mjsd

do a case for even n and a case for odd n