### Instruction: The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many of the first $20$ rows have this property? (Don't include row $0$ or row $1$). \begin{tabular}{ccccccccccc}
&&&&&1&&&&&\\
&&&&1&&1&&&&\\
&&&1&&2&&1&&&\\
&&1&&3&&3&&1&&\\
&1&&4&&6&&4&&1&\\
1&&5&&10&&10&&5&&1\\
\end{tabular}.### Response: I notice that each entry in Pascal's triangle is the sum of the two entries above it.
So, to get an even number, I need both entries above it to be either even or odd.
I also notice that the entries on the edges are always $1,$ which is odd.
So, the only way to get a row with only even numbers, except for the edges, is to have the second and second-to-last entries be even as well.
The second entry in a row is always the same as the row number, since it is the sum of $1$ and the previous row number.
So, I need the row number to be even to have a chance of getting an even second entry.
Similarly, the second-to-last entry in a row is always the same as the row number, since it is the sum of $1$ and the previous row number in reverse order.
So, I need the row number to be even to have a chance of getting an even second-to-last entry.
Therefore, I can conclude that the rows that have only even numbers, except for the edges, are the ones whose row numbers are even.
There are $10$ even numbers between $2$ and $20,$ inclusive.
So, there are $10$ rows that have this property.