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Difference between revisions of "2010 AMC 8 Problems/Problem 25"

(Created page with "==Problem== Everyday at school, Jo climbs a flight of <math>6</math> stairs. Joe can take the stairs <math>1,2</math>, or <math>3</math> at a time. For example, Jo could climb <m...")
 
(→‎Solution)
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==Solution==
 
==Solution==
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We will systematically enumerate all of the possibilities. We will denote a specific climb by using a sequence of <math>1s</math>, <math>2s</math>, and <math>3s</math> in which each sequence adds to <math>6</math>.
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We will systematically consider all of the possibilities. A valid climb can be thought of as a sequence of some or all of the numbers <math>1</math>, <math>2</math>, and <math>3</math>, in which the sum of the sequence adds to <math>6</math>. Since there is only one way to create a sequence which contains all <math>1s</math>, all <math>2s</math>, or all <math>3s</math>, there are three possible sequences which only contain one number. If we attempt to create sequences which contain one <math>2</math> and the rest <math>1s</math>, the sequence will contain two <math>2s</math> and four <math>1s</math>. We can place the <math>2</math> in either the first, second, third, fourth, or fifth position, giving a total of five possibilities. If we attempt to create sequences which contain one <math>3</math> and the rest <math>1s</math>, the sequence will contain one <math>3</math> and three <math>1s</math>. We can place the <math>3</math> in either the first, second, third, or fourth position, giving a total of four possibilities. For sequences which contain exactly two <math>2s</math> and the rest <math>1s</math>, the sequence will contain two <math>2s</math> and two <math>1s</math>. The two <math>2s</math> could be next to each other, separated by one <math>1</math> in between, or separated by two <math>1s</math> in between. We can place the two <math>2s</math> next to each other in three ways, separated by one <math>1</math> in two ways, and separated by two <math>1s</math> in only one way. This gives us a total of six ways to create a sequence which contains two <math>2s</math> and two <math>1s</math>. Note that we cannot have a sequence of only <math>2s</math> and <math>3s</math> since the sum will either be <math>5</math> or greater than <math>6</math>. We now only need to consider the case where we use all three numbers in the sequence. Since all three numbers add to <math>6</math>, the number of permutations of the three numbers is <math>3!=6</math>. Adding up the number of sequences above, we get: <math>3+5+4+6+6=24</math>. Thus, answer choice <math>\boxed{\textbf{(E)}\ 24}</math> is correct.
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Since there is only one way to create a sequence which contains all <math>1s</math>, all <math>2s</math>, or all <math>3s</math>, there are three possible sequences which only contain one number. If we attempt to create sequences which contain one <math>2</math> and the rest <math>1s</math>, the sequence will contain two <math>2s</math> and four <math>1s</math>. We can place the <math>2</math> in either the first, second, third, fourth, or fifth position, giving a total of five possibilities. If we attempt to create sequences which contain one <math>3</math> and the rest <math>1s</math>, the sequence will contain one <math>3</math> and three <math>1s</math>. We can place the <math>3</math> in either the first, second, third, or fourth position, giving a total of four possibilities. For sequences which contain exactly two <math>2s</math> and the rest <math>1s</math>, the sequence will contain two <math>2s</math> and two <math>1s</math>. The two <math>2s</math> could be next to each other, separated by one <math>1</math> in between, or separated by two <math>1s</math> in between. We can place the two <math>2s</math> next to each other in three ways, separated by one <math>1</math> in two ways, and separated by two <math>1s</math> in only one way. This gives us a total of six ways to create a sequence which contains two <math>2s</math> and two <math>1s</math>.
 
−
 
 
−
Note that we cannot have a sequence of only <math>2s</math> and <math>3s</math> since the sum will either be <math>5</math> or greater than <math>6</math>. We now only need to consider the case where we use all three numbers in the sequence. Since all three numbers add to <math>6</math>, the number of permutations of the three numbers is <math>3!=6</math>.
 
−
 
 
−
Adding up the number of sequences above, we get: <math>3+5+4+6+6=24</math>. Thus, answer choice <math>\boxed{\textbf{(E)}\ 24}</math> is correct.
 
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2010|num-b=25|after=Last Problem}}
 
{{AMC8 box|year=2010|num-b=25|after=Last Problem}}

Revision as of 20:23, 4 March 2012

Problem

Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1,2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?

$\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24$

Solution

We will systematically consider all of the possibilities. A valid climb can be thought of as a sequence of some or all of the numbers $1$, $2$, and $3$, in which the sum of the sequence adds to $6$. Since there is only one way to create a sequence which contains all $1s$, all $2s$, or all $3s$, there are three possible sequences which only contain one number. If we attempt to create sequences which contain one $2$ and the rest $1s$, the sequence will contain two $2s$ and four $1s$. We can place the $2$ in either the first, second, third, fourth, or fifth position, giving a total of five possibilities. If we attempt to create sequences which contain one $3$ and the rest $1s$, the sequence will contain one $3$ and three $1s$. We can place the $3$ in either the first, second, third, or fourth position, giving a total of four possibilities. For sequences which contain exactly two $2s$ and the rest $1s$, the sequence will contain two $2s$ and two $1s$. The two $2s$ could be next to each other, separated by one $1$ in between, or separated by two $1s$ in between. We can place the two $2s$ next to each other in three ways, separated by one $1$ in two ways, and separated by two $1s$ in only one way. This gives us a total of six ways to create a sequence which contains two $2s$ and two $1s$. Note that we cannot have a sequence of only $2s$ and $3s$ since the sum will either be $5$ or greater than $6$. We now only need to consider the case where we use all three numbers in the sequence. Since all three numbers add to $6$, the number of permutations of the three numbers is $3!=6$. Adding up the number of sequences above, we get: $3+5+4+6+6=24$. Thus, answer choice $\boxed{\textbf{(E)}\ 24}$ is correct.

See Also

2010 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 25 		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 AJHSME/AMC 8 Problems and Solutions
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