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?


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:
\[ 3 \, (\text{red}) + 1 \, (\text{green}) + 1 \, (\text{lavender}) + 3 \, (\text{yellow}) + 2 \, (\text{orange}) = 10 \text{ marbles}. \]

### 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:

\[
\binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
\]

### 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:

1. **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:
   \[
   \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21
   \]

2. **1 red marble:** Choose 1 red marble and 4 non-red marbles.
   - Number of ways to choose 1 red marble from 3:
   \[
   \binom{3}{1} = 3
   \]
   - Number of ways to choose 4 non-red marbles from 7:
   \[
   \binom{7}{4} = 35
   \]
   - Total for this case:
   \[
   3 \times 35 = 105
   \]

### Step 4: Subtract the cases with fewer than 2 red marbles.
The total number of sets with fewer than 2 red marbles is:
\[
21 \, (\text{with 0 red}) + 105 \, (\text{with 1 red}) = 126
\]

Now, subtract this from the total number of sets of 5 marbles:
\[
252 - 126 = 126
\]

### 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:

1. How many sets of five marbles include exactly two red ones?
2. What is the probability of drawing exactly three red marbles when choosing five?
3. How many sets of four marbles include no red marbles?
4. How many sets of six marbles can be made that include at least three red ones?
5. 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.

Explore how to count sets of five marbles that include at least two red marbles using combinatorics and the binomial coefficient formula. This problem involves selecting marbles from a set with specific quantities of different colors. Suitable for students in Grades 7-9.

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