Number of Odd Entries in Row of Pascal's Triangle is Power of 2

Theorem
The number of odd entries in a row of Pascal's triangle is a power of $2$.

Proof
Let $n, k \in \Z$.

Let the representations of $n$ and $k$ to the base $p$ be given by:
 * $n = 2^r a_r + \cdots + 2 a_1 + a_0$
 * $k = 2^r b_r + \cdots + 2 b_1 + b_0$

By Corollary to Lucas' Theorem:
 * $\displaystyle \dbinom n k \equiv \prod_{j \mathop = 0}^r \dbinom {a_j} {b_j} \pmod 2$

By definition, $a_j$ and $b_j$ are either $0$ or $1$.

We have:


 * $\dbinom 0 1 = 0$
 * $\dbinom 0 0 = \dbinom 1 0 = \dbinom 1 1 = 1$

Thus $\dbinom n k$ is odd none of $\dbinom {a_j} {b_j} = 0$.

We are to determine how many values of $k$ there are such that $\dbinom n k$ is odd.

We construct a $k$ such that $\dbinom n k$ is odd.

For each $a_j = 1$, we can have $b_j = 0$ or $1$.

For each $a_j = 0$, we must have $b_j = 0$.

Hence for each $a_j$, there are $2^{a_j}$ ways for the corresponding $b_j$ to be such that $\dbinom {a_j} {b_j} \ne 0$.

By Product Rule for Counting, the number of choices for $a_0, \dots, a_r$ for which $k$ is such that $\dbinom n k$ is odd is given by:


 * $\displaystyle \prod_{j \mathop = 0}^r 2^{a_j} = 2^{\sum_{j = 0}^r {a_j} }$

which is a power of $2$.

Thus the number of odd entries in a row of Pascal's triangle is a power of $2$.