Question

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

Answer 1

`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.