The ways to school
✓ With solution
A grid of streets with one stretch closed for repairs. How many shortest routes can Zaid take to school?
Every morning Zaid walks from his home to his school through a neighbourhood whose streets form a grid, as in the picture. He always takes a shortest route: along the streets, moving only up or to the right. This morning one stretch of street is closed for repairs (marked in red).
How many different shortest routes can Zaid take today?
- A12
- B14
- C16
- D20
Show the answer
Answer: B) 14. Write at each corner the number of routes that reach it. Every corner along the bottom street and along the left street can be reached in just one way. Any other corner gets the number at the corner to its left plus the number at the corner below it, except that you don't count the way in through the closed stretch. Keep adding up to the school and you get 14. Without the closure there would be 20; the closure removes the 6 routes that used it: 2 ways to reach its start, times 3 ways from its end to the school, 2 × 3 = 6.
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