From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected a

Q: Wait, why do we multiply the selection and arrangement? Can't we just arrange them all at once?

That is a great question! If you try to arrange them all at once, you might accidentally count arrangements where the dictionary is not in the middle. By separating the process into 'selection' and 'arrangement', we ensure that we only count valid configurations. We select the items first to define our set, and then we arrange that specific set according to the constraint. Multiplying these independent steps is the core of the Fundamental Counting Principle.

Q: Does the order of selecting the novels matter?

No, at the selection stage, the order does not matter. Whether you pick Novel A then Novel B, or Novel B then Novel A, you end up with the same set of 4 novels. That is why we use combinations ($^6C_4$). The order only matters when we place them on the shelf, which is why we use permutations ($4!$) in the final step.

Q: Why is the dictionary arrangement only 1 way?

The problem states the dictionary must be in the middle. Since we have already selected exactly one dictionary, and there is only one middle slot on the shelf, there is only one possible position for that dictionary. It is 'fixed' in that spot, so there is only 1 way to place it there.

Question

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is:

Options:

(A) at least 500 but less than 750

(B) at least 750 but less than 1000

(C) at least 1000

(D) less than 500

Sigma Percentile · Accepted Answer

### Analyzing the Selection Process

Before we can arrange the books, we must first curate our collection. We have a total of 6 different novels and 3 different dictionaries, and we need to select 4 novels and 1 dictionary.

Since the order of selection does not matter, we use combinations. For the novels, we choose 4 out of 6:

$$^6C_4 = \frac{6!}{4!(6-4)!} = \frac{6 	imes 5}{2 	imes 1} = 15$$

Thus, there are 15 distinct ways to choose the novels. For the dictionary, we choose 1 out of 3:

$$^3C_1 = 3$$

### The Anchor Constraint

Now, consider the shelf with 5 available slots. The constraint is absolute: the dictionary must occupy the middle position.

The 3rd slot is reserved exclusively for the dictionary. Once the dictionary is selected, there is only $1$ way to place it in the center, effectively locking it as the anchor of our display.

### The Dance of the Novels

With the dictionary fixed in the center, we are left with 4 empty slots—two to the left and two to the right. We must now place our 4 selected novels into these 4 remaining slots.

Because the order of the novels on the shelf matters, we use permutations. Arranging 4 distinct items in 4 distinct slots is calculated as:

$$4! = 4 	imes 3 	imes 2 	imes 1 = 24$$

There are 24 ways to arrange the chosen novels around the fixed dictionary.

### The Grand Finale

To find the total number of valid arrangements, we apply the **Fundamental Principle of Counting**. We multiply the number of ways to perform each independent task:

$$	ext{Total} = (	ext{Ways to select novels}) 	imes (	ext{Ways to select the dictionary}) 	imes (	ext{Ways to arrange the novels})$$

Substituting our calculated values:

$$	ext{Total} = 15 	imes 3 	imes 24$$

Breaking down the multiplication:

$$15 	imes 3 = 45$$

$$45 	imes 24 = 1080$$

The final number of valid arrangements is **1080**.

Animated Solution for Mathematics - Permutations and Combinations: From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is:

Select Answer:

Visualized Solution

Problem Setup

The Fundamental Principle

Selecting Novels

Calculating $^{6} C_{4}$

Selecting the Dictionary

Calculating $^{3} C_{1}$

The Shelf Constraint

Placing the Dictionary

Arranging the Novels

Calculating $4!$

Combining the Steps

Final Multiplication

Conclusion

The Sigma Insight: Linear Permutations

Analyzing the Selection Process

The Anchor Constraint

The Dance of the Novels

The Grand Finale

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