Mock AIME 1 Pre 2005 Problems/Problem 6

Problem

A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.

Solution

We can find a recursion. Let $D_n$ be the number of legal delivery sequences for $n$ houses. If a sequence ends with a delivery, we simply append one to $D_{n - 1}$ . If it ends in $1$ nondelivery, we append a nondelivery and a delivery to $D_{n - 2}$ . If it ends in $2$ nondeliveries, we append them and a delivery to $D_{n - 3}$ . So

$D_n = D_{n - 1} + D_{n - 2} + D_{n - 3}$ .

Thus, since clearly $D_1 = 2$ , $D_2 = 4$ , $D_3 = 7$ , we have $D_4 = 13$ , $D_5 = 24$ , $D_6 = 44$ , $D_7 = 81$ , $D_8 = 149$ , $D_9 = 274$ , $D_{10} = \boxed{504}$ .

Mock AIME 1 Pre 2005 (Problems, Source)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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Mock AIME 1 Pre 2005 Problems/Problem 6

Problem

Solution

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