 assume that the student has a cup with 11 writing implements: 5 ﻿pencils, 4 ﻿ball point pens, and 2 ﻿felt tip pens. In how many ways can the selection be made if no more than one ball point pen is selected?

We are tasked with determining the number of ways to select writing implements from a cup that contains 5 pencils, 4 ballpoint pens, and 2 felt-tip pens, given that no more than one ballpoint pen can be selected.

### Total writing implements:
- 5 pencils
- 4 ballpoint pens
- 2 felt-tip pens

We are restricted to selecting **no more than one ballpoint pen**, which means we can choose either **0** or **1** ballpoint pen. Now, let's break the problem into cases.

### Case 1: Selecting 0 ballpoint pens
In this case, we are selecting from the 5 pencils and 2 felt-tip pens, for a total of 7 implements. The number of ways to select any number of pencils and felt-tip pens is the sum of the number of subsets of these 7 items, which is \( 2^7 \). This includes selecting none, some, or all of the 7 pencils and felt-tip pens.

Thus, the number of ways to select **0 ballpoint pens** is:

\[
2^7 = 128
\]

### Case 2: Selecting 1 ballpoint pen
In this case, we select 1 ballpoint pen from the 4 available, which can be done in \( \binom{4}{1} = 4 \) ways. After selecting the ballpoint pen, we can choose any subset from the remaining 7 pencils and felt-tip pens, which again can be done in \( 2^7 = 128 \) ways.

Thus, the number of ways to select **1 ballpoint pen** is:

\[
4 \times 128 = 512
\]

### Total number of selections
The total number of ways to make the selection is the sum of the two cases:

\[
128 + 512 = 640
\]

Thus, the total number of ways to select writing implements, with the restriction that no more than one ballpoint pen is chosen, is \( \boxed{640} \).

Would you like more details or clarifications?

### 5 Related Questions:
1. How would the solution change if exactly 2 pencils must be selected?
2. What if the restriction was no more than one **felt-tip pen** instead of ballpoint pens?
3. How would the number of selections change if the total number of writing implements increased?
4. What is the total number of possible selections if no restrictions were imposed?
5. How does the inclusion-exclusion principle apply to combinatorics problems like this?

**Tip:** In combinatorics, try breaking problems into cases and using simple counting techniques to manage restrictions.

Assume that the student has a cup with 11 writing implements: 5 pencils, 4 ballpoint pens, and 2 felt-tip pens. In how many ways can the selection be made if no more than one ballpoint pen is selected?

Solve a combinatorics problem involving the selection of writing implements. Learn how to calculate the number of ways to choose pencils, ballpoint pens, and felt-tip pens, with a restriction on selecting ballpoint pens. This problem covers key concepts in subset counting and combinatorics, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation