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Math Help - Binomial coefficients

  1. #1
    Senior Member pankaj's Avatar
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    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
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Binomial Coefficients

    Hello pankaj
    Quote Originally Posted by pankaj View Post
    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
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  3. #3
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    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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