Even + Odd = Odd

For students of class 9-10 (age 14-16)
User avatar
Fatin Farhan
Posts:75
Joined:Sun Mar 17, 2013 5:19 pm
Location:Kushtia,Bangladesh.
Contact:
Even + Odd = Odd

Unread post by Fatin Farhan » Wed Feb 19, 2014 11:50 am

Let n be an odd integer greater than $$1$$. Prove that the sequence
$$ \binom{n}{1}, \binom{n}{2},...., \binom{n}{\frac{n-1}{2}}$$
contains an odd number of odd numbers.
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"

sadman sakib
Posts:17
Joined:Thu Aug 29, 2013 6:33 pm
Location:Japan Garden City, Mohammadpur, Dhaka

Re: Even + Odd = Odd

Unread post by sadman sakib » Fri Feb 21, 2014 12:18 pm

I think the problem is incomplete . Because if $n$ is an an even integer , then the sequence $ \binom{n}{1} , \binom{n}{2} , ....., \binom{n}{\frac n2}$ contains either no odd integer or an odd number of odd integer(s) . Almost similar , isn't it ?

User avatar
Labib
Posts:411
Joined:Thu Dec 09, 2010 10:58 pm
Location:Dhaka, Bangladesh.

Re: Even + Odd = Odd

Unread post by Labib » Fri Feb 21, 2014 2:09 pm

@Sakib
But the problem says that $n$ can only be an odd integer greater than $1$. Why are you bothered about even values of $n$? :-|
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.


"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes

Kiriti
Posts:27
Joined:Mon May 13, 2013 5:05 pm
Location:401/1 South Paik Para, Kalyanpur, Mirpur, Dhaka-1216

Re: Even + Odd = Odd

Unread post by Kiriti » Fri Feb 21, 2014 4:24 pm

We know, \( { n \choose k} = {n \choose n-k}\).
So, \({ n \choose 1} + { n \choose 2}+ ...... +{ n \choose \frac {n-1}{2}} = \frac {1}{2} \times [{ n \choose 1} + { n \choose 2}+ ...... + { n \choose n-1}] = \frac {1}{2} \times ( 2^n - 2 ) = 2^{n-1} - 1\)

Where, \(2^{n-1} -1\) is a odd number . That means \({ n \choose 1}, { n \choose 2}, { n \choose 3} .... { n \choose \frac {n-1}{2}} \) contains an odd numbers of odd numbers :lol:
"Education is the most powerful weapon which you can use to change the world"- Nelson Mandela

Post Reply