Difference between revisions of "2021 Fall AMC 10A Problems/Problem 18"

Line 10: Line 10:
 
==Risky, but Quick Solution==
 
==Risky, but Quick Solution==
 
The top right box has <math>4</math> choices and the top left box has <math>3</math> choices. Thus, it is reasonable to assume that the answer is a multiple of <math>12</math>. We know that the answer will not be too small or too large, so the answer is <math>\boxed{\textbf{(C)}\ 84}</math>
 
The top right box has <math>4</math> choices and the top left box has <math>3</math> choices. Thus, it is reasonable to assume that the answer is a multiple of <math>12</math>. We know that the answer will not be too small or too large, so the answer is <math>\boxed{\textbf{(C)}\ 84}</math>
 +
 +
- Aidensharp

Revision as of 18:43, 22 November 2021

Problem

A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field? [asy] draw((0,0)--(100,0)--(100,50)--(0,50)--cycle); draw((50,0)--(50,50)); draw((0,25)--(100,25)); [/asy] $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144$

Risky, but Quick Solution

The top right box has $4$ choices and the top left box has $3$ choices. Thus, it is reasonable to assume that the answer is a multiple of $12$. We know that the answer will not be too small or too large, so the answer is $\boxed{\textbf{(C)}\ 84}$

- Aidensharp