+1207 The first six rows of Pascal's triangle are shown below, beginning with row 0.
How many of the first 100 rows contain at least one even entry and no odd entries other than 1? (Rows 2 and 4 have this property, for example.)
+998 Okay. I'm gonna attempt to explain how it works.
The second and fourth rows of the pascal's triangle have only even digits (excluding the 1s).
If we look further down into Pascal's triangle, we can see that rows 8 and 16 are the next two rows with only even digits (excluding the 1).
Do you notice a pattern here?
The even digit rows in order are: \(2, 4, 8, 16...\)
The number of each subsequent row is twice the number of each preceding row.
Using this knowledge, we can find that the rows with only even numbers, less than 100 are:
\(2, 4, 8, 16, 32, 64\), which would give us an answer of \(\boxed 6 \) rows.
(P.S. When you are learning about Pasal's triangle, stuff like this is good to keep in mind)
