Difference between revisions of "2018 AMC 8 Problems/Problem 19"

Revision as of 10:27, 1 January 2023

Problem

In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?

$[asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$+$",(0,0)); draw(shift(1,0)*box); label("$-$",(1,0)); draw(shift(2,0)*box); label("$+$",(2,0)); draw(shift(3,0)*box); label("$-$",(3,0)); draw(shift(0.5,0.4)*box); label("$-$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$-$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$-$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$+$",(1,0.8)); draw(shift(2,0.8)*box); label("$+$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$+$",(1.5,1.2)); [/asy]$

$\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$

Solution 1

You could just make out all of the patterns that make the top positive. In this case, you would have the following patterns:

+−−+, −++−, −−−−, ++++, −+−+, +−+−, ++−−, −−++. There are 8 patterns and so the answer is $\boxed{\textbf{(C) } 8}$ .

-NinjaBoi2000

Solution 2

The sign of the next row on the pyramid depends on previous row. There are two options for the previous row, - or +. There are three rows to the pyramid that depend on what the top row is. Therefore, the ways you can make a + on the top is $2^3=8$ , so the answer is $\boxed{\textbf{(C) } 8}$ .

Solution 3

There is also a pretty simple approach to this problem. Since in the bottom row you can either have 4 of the same signs, 3 of the same signs and one of another, and 2 of the same signs and two of the other, this can be thought of as the 4th Row of the Pascal’s Triangle, which is $1 3 3 1$ . Since putting 3 of one sign and 1 of the other in the 4th row doesn’t work, all you need to add is $1 + 3 + 3 + 1 = 8$ , so the answer is $\boxed{\textbf{(C) } 8}$ .

Solution 4

Note that there are $2^4 = 16$ possible arrangements of the bottom row, and by symmetry, half of them result in a $+$ sign at the top and half will result in a $-$ sign at the top. Therefore, the answer is $16/2 = \boxed{\textbf{(C) }8}$ .

Video Solution

https://youtu.be/j8wm3gfOYvU

~savannahsolver

@@ Line 27: / Line 27: @@
 ==Solution 2==
-The sign of the next row on the pyramid depends on previous row.  There are two options for the previous row, - or +.  There are three rows to the pyramid that depend on what the top row is.  Therefore, the ways you can make a + on the top is <math>2^3=8</math>, so the answer is <math>\boxed{\textbf{(C) } 8}</math>
+The sign of the next row on the pyramid depends on previous row.  There are two options for the previous row, - or +.  There are three rows to the pyramid that depend on what the top row is.  Therefore, the ways you can make a + on the top is <math>2^3=8</math>, so the answer is <math>\boxed{\textbf{(C) } 8}</math>.
 ==Solution 3==

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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

Page

Toolbox

Search