Problem 82 - Algebra for JEE ADVANCED

Question

The number of ways to give 16 different things to three persons \(A, B, C\) so that \(B\) gets one more than \(A\) and \(C\) gets two more than \(B\), is a. \(\frac{16 !}{4 ! 5 ! 7 !}\) b. \(4 ! 5 ! 7 !\) c. \(\frac{16 !}{3 ! 5 ! 8 !}\) d. none of these

Step-by-Step Solution

Answer

The answer is a. \(\frac{16!}{4!5!7!}\).

Step 1: Understanding the Problem

To distribute 16 distinct items among three people, where Person B receives one more item than Person A and Person C receives two more items than Person B. This constraint sets up an equation relating the number of items each person receives.

Step 2: Formulating the Equation

Let the number of items Person A receives be denoted as \(x\). Then Person B receives \(x+1\) items and Person C receives \(x+1+2 = x+3\) items. The total number of items is 16, leading to the equation: \(x + (x+1) + (x+3) = 16\).

Step 3: Solving the Equation

Simplify and solve the equation: \(3x + 4 = 16\). Subtract 4 from both sides to get \(3x = 12\). Divide by 3 to find \(x = 4\). Therefore, Person A receives 4 items, Person B receives 5 items, and Person C receives 7 items.

Step 4: Calculating the Number of Ways

The number of ways to distribute the items can be calculated by \(\frac{16!}{4!5!7!}\). This formula represents the combination of assigning specific numbers of items to each person.

Step 5: Selecting the Correct Answer

From the calculations, the correct choice is a. \(\frac{16!}{4!5!7!}\). This matches with the calculated expression for the number of combinations based on the distribution constraints.

Key Concepts

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, dealing with the arrangement and selection of objects. When we talk about permutations, we refer to the arrangement of objects in a specific order. For example, arranging 3 books on a shelf, where the order matters, is a permutation.

In contrast, combinations focus on selecting items without concern for the order. In the case of distributing 16 different items among three people, we need to consider combinations because the order doesn't affect the distribution. Instead, what matters is the number of items each person receives.

For our problem, combinations come into play as we need to determine how many ways we can divide these 16 items so that Person A gets 4, Person B gets 5, and Person C gets 7 items. This ensures the distribution's constraints are met while considering all possible groupings.

Distribution of Items

Understanding the distribution of items requires us to follow constraints given in any scenario. Here, the challenging part is to distribute items in a way that each person's item count has a specific relation to the others.

According to the problem, Person B must receive one more item than Person A, and Person C must receive two more than Person B. To solve this distribution, we first set up an equation that represents these conditions using variables. If Person A receives 'x' items, then Person B gets 'x+1', and Person C gets 'x+3'.

These conditions lead to the total:

• Person A: x items
• Person B: x + 1 items
• Person C: x + 3 items

Simplifying these under the total of 16 items is crucial. By solving, we find that Person A receives 4 items, Person B receives 5, and Person C receives 7, effectively meeting the problem's constraints.
This type of problem demands understanding both quantitative solutions and logical reasoning to distribute items as required.

Factorial Calculations

Factorials, represented by the symbol '!', are a key part of calculating permutations and combinations. The concept of factorial is straightforward: it's the product of all positive integers up to a given number. For instance, 5! means 5 x 4 x 3 x 2 x 1, which equals 120.

In distribution problems, factorial calculations help us find how many different ways items can be arranged or grouped under specific conditions. For our exercise, the number of ways to distribute 16 items into groups of 4, 5, and 7 is calculated using the formula: \[\frac{16!}{4!5!7!}\]This formula aligns precisely with the conditions set by the problem, breaking down into choosing groups for each person with the specified item counts.

• Numerator (16!): the total number of ways to arrange all items.
• Denominators (4!, 5!, 7!): correct for overcounting by permuting items within each subgroup.

Understanding how to perform these calculations and why they are needed helps grasp the mechanics behind combinatorial distribution solutions.