Thread: Binomial coefficients

1. Binomial coefficients

Find the number of odd numbers among the following

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

2. Binomial Coefficients

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

3. 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.