A box contains 10 pens, 6 pencils, and 8 erasers, in how many ways can you select 3 writing instruments from the box, if at least one pen and any pencil toler included in the selection? О 684 954 O874 Ο 900

To solve the problem, we need to calculate the number of ways to select 3 writing instruments from the box, given that at least one pen and at least one pencil are included in the selection.

### Step 1: Understand the Problem
The box contains:
- 10 pens
- 6 pencils
- 8 erasers

We are to select 3 writing instruments with the following conditions:
1. At least one pen must be included.
2. At least one pencil must be included.

### Step 2: Consider All Possible Cases
Let's analyze the different cases based on the number of pens, pencils, and erasers:

1. **Case 1: 1 pen, 1 pencil, 1 eraser**
   - Number of ways to select 1 pen: \( \binom{10}{1} = 10 \)
   - Number of ways to select 1 pencil: \( \binom{6}{1} = 6 \)
   - Number of ways to select 1 eraser: \( \binom{8}{1} = 8 \)
   - Total for this case: \( 10 \times 6 \times 8 = 480 \)

2. **Case 2: 1 pen, 2 pencils, 0 erasers**
   - Number of ways to select 1 pen: \( \binom{10}{1} = 10 \)
   - Number of ways to select 2 pencils: \( \binom{6}{2} = 15 \)
   - Total for this case: \( 10 \times 15 \times 1 = 150 \)

3. **Case 3: 2 pens, 1 pencil, 0 erasers**
   - Number of ways to select 2 pens: \( \binom{10}{2} = 45 \)
   - Number of ways to select 1 pencil: \( \binom{6}{1} = 6 \)
   - Total for this case: \( 45 \times 6 = 270 \)

### Step 3: Add the Cases
Now, add the totals from all the cases:
\[ 480 + 150 + 270 = 900 \]

Thus, the total number of ways to select 3 writing instruments with the given conditions is **900**.

### Final Answer
The correct answer is **900**.

Let me know if you need more details or have any questions.

Here are 8 related questions you might find useful:
1. How many ways can you select 2 pencils and 1 eraser from the box?
2. What is the probability of selecting exactly 2 pens and 1 eraser?
3. How many ways can you select 3 items if there are no restrictions?
4. What if you need to select 4 writing instruments with at least one pen and pencil?
5. Can you calculate the combinations when the box contains 12 pens and 8 pencils?
6. What if you wanted at least 2 pencils in your selection of 3 writing instruments?
7. How would the answer change if the box had 5 pens instead of 10?
8. What is the difference in combinations if one pen is broken and cannot be selected?

**Tip:** Always break down combinatorial problems into cases to simplify the solution process.

Learn how to calculate the number of ways to select 3 writing instruments from a box containing pens, pencils, and erasers, ensuring at least one pen and one pencil are included. Detailed solution with combinatorial analysis and step-by-step breakdown.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation