A bag contains four red marbles, three green ones, one lavender one, two yellows, and one orange marble. HINT [See Example 7.]
How many sets of four marbles include none of the red ones?

To solve the problem, we need to determine how many sets of four marbles can be formed that do not include any red marbles. Let's break it down step by step.

### Step 1: Determine the total number of non-red marbles.
The bag contains:
- 4 red marbles (which we won't use),
- 3 green marbles,
- 1 lavender marble,
- 2 yellow marbles,
- 1 orange marble.

So, the total number of non-red marbles is:
\[ 3 \, (\text{green}) + 1 \, (\text{lavender}) + 2 \, (\text{yellow}) + 1 \, (\text{orange}) = 7 \text{ non-red marbles}. \]

### Step 2: Calculate the number of ways to choose 4 marbles from the 7 non-red marbles.
We are asked to find the number of combinations of 4 marbles that can be chosen from these 7 non-red marbles. This can be done using the combination formula:

\[
\text{Number of combinations} = \binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Where:
- \( n = 7 \) (the total number of non-red marbles),
- \( k = 4 \) (the number of marbles we want to choose).

Substituting the values:

\[
\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35
\]

### Final Answer:
There are **35** sets of four marbles that include none of the red ones.

Would you like a more detailed explanation, or do you have any other questions? Here are some related questions you might be interested in:

1. How many sets of four marbles include exactly one red marble?
2. How many sets of three marbles can be made from the non-red marbles?
3. If two marbles are drawn randomly, what's the probability that neither is red?
4. How many sets of five marbles can be made from all the marbles?
5. What is the total number of different combinations of four marbles that can be drawn from the bag?

**Tip:** When dealing with combinations, remember that the order of selection does not matter, unlike permutations where order does matter.

Learn how to calculate the number of sets of four marbles that include none of the red ones using combinatorics. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation