Counting Problems

Examples, videos, and solutions to help Grade 7 students solve counting problems related to computing percents.

New York State Common Core Math Grade 7, Module 4, Lesson 18

Worksheets for Grade 7

Lesson 18 Student Outcomes

• Students solve counting problems related to computing percents.

Lesson 18 Classwork

Opening Exercise
You are about to switch out your books from your locker but forget the order of your locker combination. You know that there are a 3, 16, and 21 in some order. What is the percent of locker combinations that start with 3?

Example 1
All of the 3-letter passwords that can be formed using the letters “A” and “B” are as follows:
AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB.
a. What percent of passwords contain at least two “B’s”?
b. What percent of passwords contain no “A’s”?

Exercises 1–2

1. How many 4-letter passwords can be formed using the letters “A” and “B”?
2. What percent of the 4-letter passwords contain
   a. no “A’s”?
   b. exactly one “A”?
   c. exactly two “A’s”?
   d. exactly three “A’s”?
   e. four “A’s”?
   f. the same number of “A’s” and “B’s”?

Example 2
In a set of 3-letter passwords, 40% of the passwords contain the letter B and two of another letter. Which of the two sets below meet the criteria? Explain how you arrived at your answer.

Exercises 3–4
3. Shana read the following problem:
“How many letter arrangements can be formed from the word “triangle” that have two vowels and two consonants (order does not matter)?”
She answered that there are 30 letter arrangements.
Twenty percent of the letter arrangements that began with a vowel actually had an English definition. How many letter arrangements that begin with a vowel have an English definition?

4. Using three different keys on a piano, a songwriter makes the beginning of his melody with three notes, C, E, and G:
CCE, EEE, EGC, GCE, CEG, GEE, CGE, GGE, EGG, EGE, GCG, EEC, ECC, ECG, GGG, GEC, CCG, CEE, CCC, GEG, CGC
a. From the list above, what is the percent of melodies with all three notes that are different?
b. From the list above, what is the percent of melodies that have three of the same notes?

Example 3
Look at the 36 points on the coordinate plane with whole number coordinates between 1 and 6, inclusive.
a. Draw a line through each of the points which have an -coordinate and -coordinate sum of 7.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 6.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 5.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 4.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 3.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 2.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 8.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 9.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 10.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 11.
Draw a line through each of the points which have an x-coordinate and y-coordinate sum of 12.
b. What percent of the 36 points have a coordinate sum of 7?
c. Write a numerical expression that could be used to determine the percent of the 36 points that have a coordinate sum of 7.
d. What percent of the 36 points have a coordinate sum of 5 or less?
e. What percent of the 36 points have a coordinate sum of 4 or 10?

• Show Step-by-step Solutions

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.