Problem 160 - A Complete Resource Book in Mathematics for JEE Main 2017

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 number of such arrangements is (A) less than 500 (B) at least 500 but less than 750 (C) at least 750 but less than 1000 (D) at least 1000

Step-by-Step Solution

Answer

The number of arrangements is 1080, so the answer is (D) at least 1000.

Step 1: Determine novel combination

First, we need to choose 4 novels out of the 6 available. We use the combination formula \( \binom{n}{r} \) to calculate the number of ways to choose 4 novels out of 6. This is given by: \[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = 15. \]

Step 2: Determine dictionary choice

Next, we choose 1 dictionary from the 3 available dictionaries. This is calculated using the combination formula \( \binom{n}{r} \) as well: \[ \binom{3}{1} = 3. \]

Step 3: Arrange novels and dictionary

The dictionary must be placed in the middle, leaving 4 positions for the novels. Thus, 4 novels can be arranged in 4 positions in \( 4! \) (factorial of 4) ways. Calculate this as: \[ 4! = 24. \]

Step 4: Calculate total arrangements

Finally, multiply the number of ways to choose novels, choose a dictionary, and arrange the novels around the dictionary: \[ \binom{6}{4} \cdot \binom{3}{1} \cdot 4! = 15 \times 3 \times 24 = 1080. \] So, the total number of arrangements is 1080.

Key Concepts

Permutation

A permutation is all about arranging objects in a specific order. Think of it as having a mix of letters and figuring out all possible ways to line them up. When the order matters, permutations come into play. For example, if you have 3 books and want to see how many ways you can arrange them on a shelf, you use permutations. This is calculated using factorials, denoted by an exclamation mark (!). For 3 books, there are 3! permutations. The formula for permutations is different if you're choosing fewer items than total available, and it looks like this:

• Given: Total items = n, Choice of items = r
• Permutation Formula: \( P(n, r) = \frac{n!}{(n-r)!} \)

Knowing permutations is essential, especially when every single order is unique, much like alphabetizing a list of words.

Combination

Combinations differ from permutations because the order doesn't matter. It is simply about selecting a group of items from a larger set. You can think of combinations as choosing toppings for a pizza. Whether you pick mushrooms then peppers or peppers then mushrooms, it's the same pizza topping combination. Here’s the combination formula:

• Choose r items from n: \( C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

For example, if you have 5 types of fruits and want to select 3 for a fruit salad, the order in which you pick them doesn't matter, only the selection does. That's a combination, helping determine the number of ways you can select certain objects.

Factorial

The concept of a factorial is simple but powerful in combinatorics. The factorial of a number n (written as n!) is the product of all positive integers up to n. It's used to calculate permutations and combinations. For example, 5! equals 5 x 4 x 3 x 2 x 1, which is 120. Factorials grow very rapidly with larger numbers. They are foundational to counting methods because they tell us how many ways we can arrange or select items. Anytime you're breaking down how many possible ways there are to arrange a list of things, you're diving into factorial territory. Understanding this principle is like having a key to decode many problems in probability and arrangements.

Probability

Probability takes center stage when calculating the likelihood of certain events. When you're asking questions like, "What's the chance of drawing an ace from a deck of cards?" or "How likely am I to win this lottery?", you're dealing with probability. It's a measure from 0 to 1, where 0 means an event won't happen, and 1 means it's a sure thing. For example, if you're tossing a coin, the probability of landing heads is 0.5. The formula for probability is:

• Probability of an event = \( \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

Probability intertwines with permutations and combinations when determining how many ways an event can occur compared to all possible events, aiding in understanding the chances of various outcomes.