This is not for help or anything; just a neat factoid. The number of odd numbers in the nth row of pascal's triangle is $\displaystyle 2^k$, where k is the the number of ones occuring in the binary representation of n.

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- Jun 24th 2010, 08:53 PMChris11Another really neat theorem!
This is not for help or anything; just a neat factoid. The number of odd numbers in the nth row of pascal's triangle is $\displaystyle 2^k$, where k is the the number of ones occuring in the binary representation of n.

- Jun 25th 2010, 12:36 AMsimplependulum
The generalization of this theorem is that if $\displaystyle n = a_m a_{m-1}... a_0 $ the representation of $\displaystyle n $ in base prime $\displaystyle p $ , then the number of the binomial coefficients which are prime to $\displaystyle p$ is $\displaystyle (a_0 + 1 )(a_1 + 1 )...(a_m + 1 ) $ .