Proving the Evenness of (\stackrel{2n}{n}) Using the Binomial Theorem

Click For Summary
SUMMARY

The discussion focuses on proving that the binomial coefficient \(\binom{2n}{n}\) is even for all integers \(n \geq 1\). Participants utilize the identity \(\binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1}\) to express \(\binom{2n}{n}\) as \(2 \cdot \binom{2n-1}{n-1}\). This transformation confirms that \(\binom{2n}{n}\) is even, as it is a product of 2 and an integer. The discussion also touches on the definition of even numbers and the properties of binomial coefficients.

PREREQUISITES
  • Understanding of binomial coefficients, specifically \(\binom{n}{k}\)
  • Familiarity with the Binomial Theorem
  • Basic knowledge of discrete mathematics
  • Induction principles in mathematical proofs
NEXT STEPS
  • Study the properties of binomial coefficients in combinatorics
  • Learn about mathematical induction techniques for proofs
  • Explore the applications of the Binomial Theorem in various mathematical contexts
  • Investigate the relationship between binomial coefficients and parity (evenness/oddness)
USEFUL FOR

This discussion is beneficial for students in discrete mathematics, mathematicians focusing on combinatorics, and anyone interested in understanding the properties of binomial coefficients and their proofs.

ipitydatfu
Messages
13
Reaction score
0

Homework Statement



prove that ([tex]\stackrel{2n}{n}[/tex]) is even when n [tex]\geq1[/tex]

Homework Equations



as a hint they gave me this identity:
[tex]\stackrel{n}{k}[/tex]= (n/k)([tex]\stackrel{n-1}{k-1}[/tex])

The Attempt at a Solution



by using that identity i got:

([tex]\stackrel{2n}{n}[/tex]) = (2n/n) ([tex]\stackrel{2n-1}{n-1}[/tex])
= (2) ([tex]\stackrel{2n-1}{n-1}[/tex])

i thought anything multiplied by 2 is an even number. but then again this is discrete math. how would i inductively show that this is true?
 
Physics news on Phys.org
That's pretty much it. The definition of "even number" is that it is of the form 2k for some integer k. Do you already know that [itex]\left(\begin{array}{c}n\\2i\end{array}\right)[/itex] is always an integer?
 
oh yeah! I forgot about that! thanks!
 

Similar threads

Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4
  • · Replies 12 ·
Replies
12
Views
2K
If p is even and q is odd, then ##\binom{p}{q}## is even
  • · Replies 19 ·
Replies
19
Views
4K
Probability for a binomially distributed variable X
  • · Replies 7 ·
Replies
7
Views
2K
Show that the limit (1+z/n)^n=e^z holds
  • · Replies 6 ·
Replies
6
Views
3K
What Determines the Values of Legendre Polynomials at Zero?
  • · Replies 2 ·
Replies
2
Views
3K
Prove the equation using Newton's binom
  • · Replies 17 ·
Replies
17
Views
2K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
Find the limit using Riemann sum
  • · Replies 4 ·
Replies
4
Views
2K
Probability - Random Variables
  • · Replies 3 ·
Replies
3
Views
3K
How Can We Prove a Summation Identity by Double Counting?
  • · Replies 18 ·
Replies
18
Views
2K