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hw7sol - Homework 7 Solutions 1 In walking from work back...

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Homework 7 Solutions 1. In walking from work back home at the end of the day, Grigor must travel six blocks east and four blocks south. How many possible routes can he take, assuming that he goes either east or south at each intersection? How many routes are there if he needs to stop by the dry-cleaners on the way home, which is located at the center of the grid (that is, two blocks south and three east of his workplace)? Grigor will travel a total of 10 blocks, of which any 6 can be going east. Once we pick the 6 blocks on which he will go east, we’ve completely specified his route, as he must go south on the other 4. Thus there are 10 6 = 10 4 routes. For the second part, we split his route into two parts. First, he must go from his work to the dry-cleaners. By the same logic as before, there are ( 5 3 ) such routes. Then he must go from the dry-cleaners to his home; again there are ( 5 3 ) ways for him to do so. Since there are ( 5 3 ) ways to choose the first part of his journey and ( 5 3 ) ways to choose the second part, the multiplication principle tells us that there are 5 3 2 = 5 2 2 total routes.
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