Math Problem Statement
A bag contains three red marbles, one green one, one lavender one, three yellows, and two orange marbles. HINT [See Example 7.]
How many sets of five marbles include at least two red ones?
Solution
To solve the problem of finding how many sets of five marbles include at least two red ones, we'll take the following approach:
Step 1: Identify the total number of marbles and the red marbles.
The bag contains:
- 3 red marbles,
- 1 green marble,
- 1 lavender marble,
- 3 yellow marbles,
- 2 orange marbles.
So, the total number of marbles is:
Step 2: Calculate the total number of ways to select 5 marbles.
First, we calculate the total number of ways to choose 5 marbles out of the 10 available:
Step 3: Calculate the number of ways to select 5 marbles with fewer than 2 red ones.
We must now subtract the number of combinations that have fewer than 2 red marbles. These cases are:
-
0 red marbles: All 5 marbles are selected from the non-red marbles.
- Number of non-red marbles = 7 (1 green + 1 lavender + 3 yellow + 2 orange)
- Combinations:
-
1 red marble: Choose 1 red marble and 4 non-red marbles.
- Number of ways to choose 1 red marble from 3:
- Number of ways to choose 4 non-red marbles from 7:
- Total for this case:
Step 4: Subtract the cases with fewer than 2 red marbles.
The total number of sets with fewer than 2 red marbles is:
Now, subtract this from the total number of sets of 5 marbles:
Final Answer:
There are 126 sets of five marbles that include at least two red ones.
Would you like more details, or do you have any other questions? Here are some related questions:
- How many sets of five marbles include exactly two red ones?
- What is the probability of drawing exactly three red marbles when choosing five?
- How many sets of four marbles include no red marbles?
- How many sets of six marbles can be made that include at least three red ones?
- How many sets of five marbles include at least one yellow marble?
Tip: When solving problems with "at least" conditions, it's often helpful to use complementary counting—counting what you don't want and subtracting from the total.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Combinatorics
Probability
Formulas
Binomial coefficient formula
Theorems
-
Suitable Grade Level
Grades 7-9
Related Recommendation
Combinatorics Problem: Finding Sets of Five Marbles with at Most One Yellow Marble
Combinatorics: Sets of Marbles with All Red Marbles Included
Combinatorics: Calculating Sets with Specific Marbles
How to Find Sets of Five Marbles Including at Most One Yellow Marble
Counting Sets of Marbles with Lavender or Yellow: A Combinatorics Problem