# Problem of the Day

Gerald and his family are at point when they receive an alert that a cloud of poisonous gas is moving toward them from the southwest. They need to avoid the cloud of gas by driving to a shelter at point . If they must stay on the grid and are only able to drive north and east, how many different routes can they take from point to point ?

A. 12

B. 15

C. 21

D. 30

E. 35

### Solution

[latexpage]

Gerald and his family need to go north three times and east four times to get to the shelter. They have to make a sequence of seven moves. We can represent each sequence by a string of three N’s and four E’s. For example, one sequence could be NNNEEEE, where they immediately go north three times, followed by going east four times. To determine how many possible routes there are, we need to determine how many unique arrangements there are of the letters NNNEEEE. To do this, picture seven blank spots that need to be filled in with the three N’s and four E’s: _ _ _ _ _ _ _. Once you decide where the four E’s go, the three N’s are filled in automatically, because there will only be three spots left. You can choose the spots for the E’s in $7 \choose 4$ ways. ${7\choose 4}=\frac{7!}{3!4!}=35$.

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