Question

From 6 different novels and 3 different dictionaries 4 novels and, 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the no, of such arrangements is (a) Less then 500 (b) At least 500 but less then 750 (c) At least 750 but less then 1000 (d) At least 1000

Short answer

The number of such arrangements is at least 1000.

Step-by-step solution

Selecting the items

We need to select 4 novels out of 6 and 1 dictionary out of 3. There are 6C4 ways to select 4 novels and 3C1 ways to select 1 dictionary. The total number of possible selections is the product of these values: \( ^6C_4 \times ^3C_1 = \frac{6!}{4!2!}\times \frac{3!}{1!2!} \)

Calculating the number of selections

Now, let's calculate the values of the factorials: \( ^6C_4 \times ^3C_1 = \frac{6 \times 5 \times 4!}{4!\times 2 \times 1} \times \frac{3 \times 2!}{1!\times 2!} \) We can cancel out the common terms: \( ^6C_4 \times ^3C_1 = \frac{6 \times 5}{2} \times 3 = 3 \times 5 \times 3 = 45 \) So there are 45 possible selections of 4 novels and 1 dictionary.

Arranging the selected items

As the dictionary should always be in the middle, there is only 1 possible position for the dictionary. The 4 selected novels can be arranged in 4! different ways. Therefore, the total number of arrangements is: Total arrangements = Number of selections × Possible arrangements of novels Total arrangements = 45 × 4!

Calculating the number of arrangements

Let's find the value of 4!: 4! = 4 × 3 × 2 × 1 = 24 Now, we can find the total arrangements: Total arrangements = 45 × 24 = 1080 As per the given options, the number of arrangements is: (d) At least 1000

Key concepts

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, a field of mathematics that deals with counting and arranging objects. While they both relate to the selection of items, they're used in different ways.

- **Permutations** are about finding the number of ways to arrange a set of objects where the order matters. For instance, arranging books on a shelf is an example of permutations since the order affects the result.
- **Combinations** are used when the order does not matter, such as selecting a group of people for a team. This exercise deals primarily with combinations when selecting the novels and permutations when arranging them.

In our problem, we calculate the combinations of selecting 4 novels from a set of 6, denoted as \(^6C_4\), and 1 dictionary from 3, denoted as \(^3C_1\). Combining these methods, we first determine the possible sets of novels and dictionaries, and then use permutations to understand how these can be arranged on the shelf.

Factorials

Factorials denote a series of multiplications from a given number down to 1. It's a crucial concept in both permutations and combinations. Mathematically, the factorial of a number \(n\) is expressed as \(n!\), which equals \(n \times (n-1) \times (n-2) \times \ldots \times 1\).

For instance, in this exercise, the number of ways to arrange the novels depends on factorials. We calculated \(4!\) when arranging 4 selected novels, which equaled \(4 \times 3 \times 2 \times 1 = 24\). This represents all possible orders these novels can be arranged in real life.

In combination problems such as selecting novels (\(^6C_4\)), factorials help simplify the calculations by canceling repeated terms, making it easier to understand how we arrive at the number of possible selections or arrangements. Understanding factorials is essential for tackling permutation and combination problems efficiently.

Arrangements

Arrangements refer to the different ways to order items, especially important when conditions or restrictions are involved, like in our exercise, where a dictionary must occupy the middle position on a shelf.

- **Positioning the Dictionary**: Here, the dictionary has a specific spot it should occupy, reducing its arrangement to just one possibility, emphasizing how arrangements can be controlled by external conditions.
- **Arranging the Novels**: Once the dictionary is fixed, the novels can be rearranged. With 4 novels, there are \(4!\) or 24 possibilities due to different potential orders amongst them.

Combining the calculation of combinations (choosing which novels and dictionary) with these potential arrangements gives us the total count of how books can be organized on a shelf, which in this case results in 1080 different scenarios as dictated by the fixed condition for the dictionary.