Exercise 1.9

[label=]Each path from $(0,0)$ to $(110,111)$ can be understood as a sequence of $R$’s and $U$’s, where the first represents a step to the right and the latter a step up. But since the grid is 110 squares tall by 111 squares wide, then we must have 110 steps to the right and 111 steps up. So we note the problem is equivalent of calcultaing the total number of strings 221 characters long containing 110 $R$’s and 111 $U$’s. But that is

$$\frac{221!}{110!111!}.$$

There are $\frac{221!}{110!111!}$ ways to move from $(0,0)$ to $(110,111)$, and $\frac{200!}{100!100!}$ ways to move from $(110,111)$ to $(210,211)$. So, there are

$$\left(\frac{221!}{110!111!}\right)\left(\frac{200!}{100!100!}\right)$$

ways to go from $(0,0)$ to $(210,211)$.