Difference between revisions of "2019 AMC 10B Problems/Problem 25"

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-Solution by MagentaCobra
 
-Solution by MagentaCobra
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2) After any given zero, the next zero must appear exactly two or three spots down the line. And we started at position 1 and ended at position 19, so we moved over 18. Therefore, we must add a series of 2's and 3's to get 18. How can we do this?
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Option 1: nine 2's (there is only 1 way to arrange this).
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Option 2: two 3's and six 2's (there are 8 choose 2 ways to arrange this, or 28).
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Option 3: four 3's and three 2's (there are 7 choose 3 ways to arrange this, or 35).
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Option 4: six 3's (there is only 1 way to arrange this).
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Sum the four numbers given above: 1+28+35+1=65 \quad \boxed{C}$
  
 
==See Also==
 
==See Also==

Revision as of 14:42, 14 February 2019

The following problem is from both the 2019 AMC 10B #25 and 2019 AMC 12B #23, so both problems redirect to this page.

Problem

How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?

$\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$

Solution

We can deduce that any valid sequence of length $n$ wil start with a 0 followed by either "10" or "110". Because of this, we can define a recursive function:

$f(n) = f(n-3) + f(n-2)$

This is because for any valid sequence of length $n$, you can remove either the last two numbers ("10") or the last three numbers ("110") and the sequence would still satisfy the given conditions.

Since $f(5) = 1$ and $f(6) = 2$, you follow the recursion up until $f(19) = 65 \quad \boxed{C}$

-Solution by MagentaCobra


2) After any given zero, the next zero must appear exactly two or three spots down the line. And we started at position 1 and ended at position 19, so we moved over 18. Therefore, we must add a series of 2's and 3's to get 18. How can we do this? Option 1: nine 2's (there is only 1 way to arrange this). Option 2: two 3's and six 2's (there are 8 choose 2 ways to arrange this, or 28). Option 3: four 3's and three 2's (there are 7 choose 3 ways to arrange this, or 35). Option 4: six 3's (there is only 1 way to arrange this).

Sum the four numbers given above: 1+28+35+1=65 \quad \boxed{C}$

See Also

2019 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Last Problem
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2019 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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