Ways to write the word mathematics from a grid In how many ways can you read the word mathematics in this grid, starting form the top left, ending in the bottom right?
$MATHEMATI\\
ATHEMATIC\\
THEMATICS$
 A: There is $1$ way to reach the any of the top row of letters, and $1$ to reach any of the letters in the leftmost column.
For any of the other letters, they can either be reached from the letter directly above, or the letter directly to the left. Therefore, 

The number of ways to reach a letter in the grid is equal to the number of ways to reach the letter above it, plus the number of ways to reach the letter to the left of it.

Applying this rule allows us to fill out the grid quite quickly:

This means that the number of ways to reach the bottom right corner is $\boxed{45\,}$.

You may notice that these numbers are very similar to those appearing in Pascal's triangle. This is not a coincidence. The reason is that the rule of adding the number of ways to reach the letter above to the number of ways to reach the letter to the left, is the analogue of Pascal's rule. Pascal's triangle can be generated from an initial row of all zeroes and a $1$, just as your grid can be generated from an initial diagonal containing the top left corner and zeroes for the rest of the entries. 

You could also uniquely determine the path taken to the bottom right corner by choosing the two positions where you take a step down, instead of a step to the right. 
Your path has $10$ steps, one for each letter in "mathematics" after the initial "m". $2$ of these steps must be steps down, and the other $8$ are steps to the right. 
This means that the number of paths is the number of ways to choose $2$ objects from a set of $10$, which is the binomial coefficient
$$\binom{10}{2} = \frac{10 \cdot 9}{2} = 45$$
In fact, the entries of Pascal's triangle are exactly these binomial coefficients. 
A: HINT
Show that if $i$ is the number of the row, numbered $0$ through $2$, and $j$ the number of the column, numbered $1$ through $10$, then the number of ways to get from your starting point $(0,1)$ to the point $(i,j)$ is $j \choose i$
