Grades 5-6 Video Solutions 2023-2023_1-2

Video Transcription

Question 16. The diagram shows five rectangles. Lucas wants to color the rectangles red, blue, and yellow so that any two adjacent  rectangles are colored different colors.  In how many different ways can he do this?  Let's start off by drawing our five rectangles.  And picking the center rectangle to be blue.  This one is blue. Let's make this next one red. And this makes us have this one be yellow. And then we have to make these yellow and red.  Now, alternatively, we could switch all the yellows and reds and still have a valid colored rectangle set up.  Now we have two different options, with yellow and red, or red and yellow.  However, since we have three different colors, instead of blue, we could have made it yellow. And then all of the yellow would be blue.  And then we could also have, instead of red, had blue, and instead of blue, had red.  So since there are three colors and two options, our answer will be D, six.

Video Summary

Lucas can color the five rectangles in 6 different ways, ensuring adjacent rectangles have different colors. By choosing the center rectangle's color first and alternating the surrounding colors, two main combinations appear: when the center is blue, the adjacent ones can be red-yellow or yellow-red. These patterns repeat by swapping colors, considering three options for the center (red, blue, or yellow). Ultimately, with these configurations, there are six distinct ways to color the rectangles.

Keywords

rectangles

coloring

patterns

combinations

distinct