How many odd numbers are IN the 100th row of pascals triangle?

1 Answer
George C.
Jul 14, 2018

#8#

Explanation:

There is an interesting property of Pascal's triangle that the #n#th row contains #2^k# odd numbers, where #k# is the number of #1#'s in the binary representation of #n#.

Note that the #n#th row here is using a popular convention that the top row of Pascal's triangle is row #0#. This is not my preferred convention, but has some nice properties:

  • The #n#th row contains the coefficients of the expansion of #(a+b)^n#.

  • The sum of the #n#th row is #2^n#

So if we follow the popular convention, then the "#100#th row" will contain #2^k# odd numbers where #k# is the number of #1#'s in the binary representation of #100#:

#100 = 64 + 32 + 4 = 2^6+2^5+2^2 = 1100100_2#

So #k=3# and the number of terms in the #100#th row that are odd is #2^3 = 8#.

If I have time, I may add a proof of this interesting property.

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