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Chapter 5: Problem 413

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

Expert verified
The number of such arrangements is at least 1000.

Step by step solution

01

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: \( ^6C_4 \times ^3C_1 = \frac{6!}{4!2!}\times \frac{3!}{1!2!} \)
02

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.
03

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!
04

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

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