Please solve questions 3 and 5. **Problem on Counting Arrangements of Identical Items**In this combinatorial problem, there are three identical yellow books, two identical black books, and four identical orange books. The goal is to determine how many different patterns are possible if these books are arranged on a shelf.**Question:**How many different patterns are possible if the books are arranged on a shelf?**Options:**- [A] 24- [B] 362,880- [C] 9- [D] 1260**Solution Explanation:**This problem is related to counting permutations of multiset. The formula to find the number of different arrangements (patterns) when there are groups of identical items is:\[ \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]Here, $n$ is the total number of items, and $n_1, n_2, n_3$, etc., are the counts of each type of identical items.In our problem:- Total number of books $n = 3 + 2 + 4 = 9$- Number of yellow books $n_1 = 3$- Number of black books $n_2 = 2$- Number of orange books $n_3 = 4$Plugging these values into the formula:\[ \frac{9!}{3! \cdot 2! \cdot 4!} \]Calculating the factorials:- $9! = 362880$- $3! = 6$- $2! = 2$- $4! = 24$So,\[ \frac{362880}{6 \cdot 2 \cdot 24} = \frac{362880}{288} = 1260 \]Thus, the correct answer is:- [D] 1260 **Finding the Equation of the Perpendicular Bisector**The question requires us to find the equation of the perpendicular bisector of the line segment connecting the points $(0, 2)$ and $(-6, -8)$.Options are provided as follows:**[A]** \[y = \frac{3}{5}x - \frac{6}{5}\]**[B]** \[y = -\frac{3}{5}x - \frac{24}{5}\]**[C]** \[y = \frac{3}{5}x + 2\]**[D]** \[y = -\frac{3}{5}x + 2\]### Steps to Solve:1. **Find the midpoint** of the segment connecting $(0, 2)$ and $(-6, -8)$: \[ \left( \frac{0 + (-6)}{2}, \frac{2 + (-8)}{2} \right) = \left( \frac{-6}{2}, \frac{-6}{2} \right) = (-3, -3) \]2. **Determine the slope** of the line segment: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-8 - 2}{-6 - 0} = \frac{-10}{-6} = \frac{5}{3} \]3. **Find the slope of the perpendicular bisector** (negative reciprocal of the original slope): \[ \text{perpendicular slope} = -\frac{3}{5} \]4. **Use the point-slope form of the equation** of a line with the midpoint $(-3, -3)$ and slope $-\frac{3}{5}$: \[ y - y_1 = m(x - x_1) \] \[ y - (-3) = -\frac{3}{5}(x - (-3)) \] Simplify the equation above to match one of the given options.5. **Match the simplified equation** to one of the provided choices to find the correct answer.This will help determine which of the provided options (

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There are three identical yellow books, two identical black books, and four identical orange books. How many different patterns are possible if the books are arranged on a shelf? [A] 24 [B] 362,880 [C] 9 [D] 1260

There are three identical yellow books, two identical black books, and four identical orange books. How many different patterns are possible if the books are arranged on a shelf? [A] 24 [B] 362,880 [C] 9 [D] 1260

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.5: Counting Principles

Problem 46SE: How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of 8 freshmen and...

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Concept explainers

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.

Topic Video

Question

Please solve questions 3 and 5.

**Problem on Counting Arrangements of Identical Items**

In this combinatorial problem, there are three identical yellow books, two identical black books, and four identical orange books. The goal is to determine how many different patterns are possible if these books are arranged on a shelf.

**Question:**
How many different patterns are possible if the books are arranged on a shelf?

**Options:**
- [A] 24
- [B] 362,880
- [C] 9
- [D] 1260

**Solution Explanation:**
This problem is related to counting permutations of multiset. The formula to find the number of different arrangements (patterns) when there are groups of identical items is:

\[ \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]

Here, $n$ is the total number of items, and $n_1, n_2, n_3$, etc., are the counts of each type of identical items.

In our problem:
- Total number of books $n = 3 + 2 + 4 = 9$
- Number of yellow books $n_1 = 3$
- Number of black books $n_2 = 2$
- Number of orange books $n_3 = 4$

Plugging these values into the formula:

\[ \frac{9!}{3! \cdot 2! \cdot 4!} \]

Calculating the factorials:
- $9! = 362880$
- $3! = 6$
- $2! = 2$
- $4! = 24$

So,

\[ \frac{362880}{6 \cdot 2 \cdot 24} = \frac{362880}{288} = 1260 \]

Thus, the correct answer is:
- [D] 1260

**Finding the Equation of the Perpendicular Bisector**

The question requires us to find the equation of the perpendicular bisector of the line segment connecting the points $(0, 2)$ and $(-6, -8)$.

Options are provided as follows:

**[A]** \[
y = \frac{3}{5}x - \frac{6}{5}
\]

**[B]** \[
y = -\frac{3}{5}x - \frac{24}{5}
\]

**[C]** \[
y = \frac{3}{5}x + 2
\]

**[D]** \[
y = -\frac{3}{5}x + 2
\]

### Steps to Solve:

1. **Find the midpoint** of the segment connecting $(0, 2)$ and $(-6, -8)$:

\[
\left( \frac{0 + (-6)}{2}, \frac{2 + (-8)}{2} \right) = \left( \frac{-6}{2}, \frac{-6}{2} \right) = (-3, -3)
\]

2. **Determine the slope** of the line segment:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-8 - 2}{-6 - 0} = \frac{-10}{-6} = \frac{5}{3}
\]

3. **Find the slope of the perpendicular bisector** (negative reciprocal of the original slope):

\[
\text{perpendicular slope} = -\frac{3}{5}
\]

4. **Use the point-slope form of the equation** of a line with the midpoint $(-3, -3)$ and slope $-\frac{3}{5}$:

\[
y - y_1 = m(x - x_1)
\]

\[
y - (-3) = -\frac{3}{5}(x - (-3))
\]

Simplify the equation above to match one of the given options.

5. **Match the simplified equation** to one of the provided choices to find the correct answer.

This will help determine which of the provided options (

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