Binomial coefficients

**pankaj** · February 13th 2009, 07:08 AM

Find the number of odd numbers among the following

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

**Grandad** · February 13th 2009, 08:32 AM

Hello pankaj

Originally Posted by pankaj

Find the number of odd numbers among the following

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

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.

Grandad

**Soroban** · February 13th 2009, 12:22 PM

Hello, pankaj!

Find the number of odd numbers among the following:

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

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 $O$ ).

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.

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