# Binomial coefficients

• Feb 13th 2009, 07:08 AM
pankaj
Binomial coefficients
Find the number of odd numbers among the following

$\displaystyle \binom{n}{0}$,$\displaystyle \binom{n}{1},$$\displaystyle \binom{n}{2},........$$\displaystyle \binom{n}{n}.$
where $\displaystyle n\in N$
• Feb 13th 2009, 08:32 AM
Binomial Coefficients
Hello pankaj
Quote:

Originally Posted by pankaj
Find the number of odd numbers among the following

$\displaystyle \binom{n}{0}$,$\displaystyle \binom{n}{1},$$\displaystyle \binom{n}{2},........$$\displaystyle \binom{n}{n}.$
where $\displaystyle n\in N$

I don't know if this will be of any help, but I am attaching an Excel spreadsheet (which I hope you can read) showing the results for n = 0 to 32. The first sheet (formulae) uses the ISODD function (which is part of the Analysis Tool Pack); the second sheet just shows the results.

There are some patterns here, but at present I can't really see what to do next.

• Feb 13th 2009, 12:22 PM
Soroban
Hello, pankaj!

Quote:

Find the number of odd numbers among the following:

. . $\displaystyle \binom{n}{0},\;\binom{n}{1},\;\binom{n}{2}\; \hdots\;\binom{n}{n}\qquad \text{ where }n\in N$

I don't have a formula yet, but look at the pattern . . .

We are dealing with the numbers in Pascal's Triangle.
Note the locations of the odd numbers (denoted $\displaystyle O$).
Code:

                              O                             O  O                           O  .  O                         O  O  O  O                       O  .  .  .  O                     O  O  .  .  O  O                   O  .  O  .  O  .  O                 O  O  O  O  O  O  O  O               O  .  .  .  .  .  .  .  O             O  O  .  .  .  .  .  .  O  O           O  .  O  .  .  .  .  .  O  .  O         O  O  O  O  .  .  .  .  O  O  O  O       O  .  .  .  O  .  .  .  O  .  .  .  O     O  O  .  .  O  O  .  .  O  O  .  .  O  O   O  .  O  .  O  .  O  .  O  .  O  .  O  .  O O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O

And this triangular pattern continues down Pascal's Triangle.