Problem 109 - Algebra for JEE ADVANCED

Question

Messages are conveyed by arranging four white, one blue and three red flags on a pole. Flags of the same colour are alike. If a message is transmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is a. 45 b. 65 c. 125 d. 185

Step-by-Step Solution

Answer

The total number of transmitted messages is 125.

Step 1: Understand the Components

There are total 8 flags: 4 white, 1 blue, and 3 red flags. We need to choose exactly 6 flags for arranging on a pole.

Step 2: Categorize the Choices

We have three different categories to choose from: all white, blue, and red flags. We need to choose how many flags from each color will be used in 6 positions. Here are some possible combinations: - 4 white, 1 blue, 1 red - 4 white, 2 red - 3 white, 1 blue, 2 red - 3 white, 3 red

Step 3: Calculate Arrangements for Each Combination

For each chosen combination, calculate the permutations taking into account identical flags.**Combination 1:** 4 white, 1 blue, 1 red - Total flags = 6 - Formula = \( \frac{6!}{4! \, 1! \, 1!} \) - Calculation = \( \frac{720}{24 \, 1 \, 1} = 30 \)**Combination 2:** 4 white, 2 red - Total flags = 6 - Formula = \( \frac{6!}{4! \, 2!} \) - Calculation = \( \frac{720}{24 \, 2} = 15 \)**Combination 3:** 3 white, 1 blue, 2 red - Total flags = 6 - Formula = \( \frac{6!}{3! \, 1! \, 2!} \) - Calculation = \( \frac{720}{6 \, 1 \, 2} = 60 \)**Combination 4:** 3 white, 3 red - Total flags = 6 - Formula = \( \frac{6!}{3! \, 3!} \) - Calculation = \( \frac{720}{6 \, 6} = 20 \)

Step 4: Sum the Arrangements

Add up the number of arrangements from each of the combinations. - 30 (from 4W, 1B, 1R) - 15 (from 4W, 2R) - 60 (from 3W, 1B, 2R) - 20 (from 3W, 3R) Total = 30 + 15 + 60 + 20 = 125

Key Concepts

Combinatorics

Combinatorics is all about counting. It's a branch of mathematics focusing on how to count and arrange things efficiently. When you have a set of items, in this case, flags, and you want to know how many different ways they can be organized, you dive into the world of combinatorics. For example, you can have different combinations of flag colors to create different messages. When working with the combinatorics of flag arrangements, you align which flags to choose and in what sequence, leading to distinct messages. The crucial part is defining all possible categories based on how many flags you use from each color. Instead of merely listing possibilities, we use strategies to systematically count unique arrangements based on color distinctions.

Factorial Calculations

Factorial calculations are vital in permutations, as they determine how many ways a set of items can be ordered. The factorial of a number, written as \(!\), is the product of all positive integers up to that number. For example, 6! means 6 times 5 times 4 times 3 times 2 times 1, which equals 720. In permutation problems with repetitive items, like flags of identical colors, we adjust this factorial calculation. Given repetitions, the formula divides the total permutation count by factorials of repeated items. In this exercise, the permutations consider flags identical in color, using formulas such as \( \frac{6!}{4! \cdot 1! \cdot 1!} \). This adjusts the permutations by accounting for non-distinct arrangements caused by identical flag colors.

Unique Arrangements

Unique arrangements refer to the different ways a set of objects can be organized considering existing restrictions, like identical items and chosen numbers. In problems like arranging flags of different colors, uniqueness comes from determining how diverse each arrangement can be within the provided constraints. For instance, using 6 flags from a set with repeats, there are multiple unique combinations, such as 4 white and 2 red, or 3 white, 1 blue, and 2 red. Calculating these involves considering how the flags' repetition and position choices affect distinct sequences. This leads to correctly counting messages or distinct sequences from the flags. Mathematics aids by forming solutions where combinations and permutations highlight distinct sequences while adjusting for identical flags’ influence.