36Got it.....! :) :)
Let a1, a2, ........, an be an arbitrary permutation of the numbers 1,2,...........,n, where n is an
odd number. Prove that the product.
(a1-1)(a2-2)................(an-n) is even
Help?.................
8 Answers
62it means a number formed by rearranging the digits of the original
eg 123 is a permutation of 231
36couldn't understand.....
if 123 is a permutation of 231 then
can 321 be a permutation of 231?
3662this one is simple..
see the thing is that look at
(a1-1)(a3-3)................(an-n) (only the odd terms)
if we can prove that this is even we are done..
now there are (n+1)/2 odd numbers from 1 to n and (n-1)/2 even numbers...
Which means the number of odd numbers is 1 more than the number of even numbers..
Now revisit (a1-1)(a3-3)................(an-n) (only the odd terms)
If the product is odd, each of a1, a3, a5 are even..
Which means there are as many even numbers as there are odd numbers
which is a contradication...
36Out of memory.......
1st confusion : How are there (n+1)/2 odd numbers and (n-1)/2 even numbers from from 1 to n
please explain in a bit better way...
This is maybe not my cup of tea...............!
36Sir first of all i didn't understand the question.....
But the confusion which i asked you is clear now...
As from 1 to n, if, n is odd then there are (n + 1)/2 and (n - 1)/2 even numbers
and if , n is even then there there are n/2 even as well as odd numbers.....
got that......... :) :) :) :)
But what does the question say?...
How many permutation of 1 are there?... maybe only 1...
11Permutation means each of the different arrangements which can be made by taking some or all of a number of given things or objects at a time.
In permutation order of appearance of things is taken into account.