Problem 102 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of ordered triplets of positive integers which are solutions of the equation \(x+y+z=100\) is (A) 5081 (B) 6005 (C) 4851 (D) None of these

Step-by-Step Solution

Answer

4851; the answer is (C).

Step 1: Understanding the Equation

We are given the equation \(x + y + z = 100\) and we need to find the number of ordered triplets of positive integers \((x, y, z)\) that satisfy this equation. Each variable must take a positive integer value.

Step 2: Reformulation Using Non-Negative Integers

The task requires positive integers, so we can rewrite \(x = x' + 1, y = y' + 1, z = z' + 1\) where \(x', y', z'\) are non-negative integers. Substituting in the equation, we get \(x' + 1 + y' + 1 + z' + 1 = 100\). This simplifies to \(x' + y' + z' = 97\). Now, we solve for non-negative integer solutions to this new equation.

Step 3: Applying the Stars and Bars Theorem

To count non-negative solutions of \(x' + y' + z' = 97\), we can apply the "Stars and Bars" theorem. According to this theorem, the number of solutions in non-negative integers is given by the formula \(\binom{n+k-1}{k-1}\), where \(n\) is the total to be obtained (97 in this case) and \(k\) is the number of variables (3 here).

Step 4: Calculating the Number of Solutions

Plugging values into the formula, we have: \(\binom{97 + 3 - 1}{3 - 1} = \binom{99}{2}\). To calculate \(\binom{99}{2}\), compute: \[ \binom{99}{2} = \frac{99 \times 98}{2} = 4851 \]

Step 5: Identifying the Correct Answer

The computed number of ordered triplets is 4851. Comparing this to the options given, the answer is (C) 4851.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns or structures in sets. It plays a crucial role in solving problems where we need to find the number of ways something can happen. This includes determining combinations, permutations, and understanding how systems can be organized.
In our exercise, combinatorics is used to find the number of solutions to the equation \(x + y + z = 100\). The task involves counting different ways to distribute a total of 100 units among three variables—\(x, y,\) and \(z\).
Combinatorics is versatile and is commonly applied to problems in probability, statistics, computer science, and many other fields. Understanding the basic tools in combinatorics, such as the stars and bars theorem, is essential for solving problems involving non-negative solutions and ordered arrangements.

Ordered Triplets

An ordered triplet, in mathematics, is simply a group of three numbers or variables arranged in a specific sequence. These are denoted as \((x, y, z)\) where each position in the triplet is important and switching the order produces a different triplet.
Ordered triplets are crucial in problems where the arrangement matters, as different orderings represent different solutions. For example, the triplet \((3, 4, 5)\) is different from \((5, 4, 3)\).
In the exercise given, finding ordered triplets of positive integers means calculating all possible sequences of values for \(x\), \(y\), and \(z\) that add up to a certain number, while each value remains a positive integer.

• The importance of ordered triplets is highlighted when calculating permutations and combinations where order impacts the outcome.
• In our solution, the focus was on the arrangement of values satisfying the equation while maintaining their order.

Non-negative Solutions

Non-negative solutions refer to sets of numbers that all are either zero or positive. Unlike positive solutions, non-negative solutions allow for zero as a valid entry.
For example, if solving the equation \(a + b = 5\) for positive integer solutions, both \(a\) and \(b\) must be greater than zero. However, for non-negative solutions, \(a\) or \(b\) could be zero.
In the provided problem, we transformed the original equation into a form allowing us to use non-negative numbers: \(x' + y' + z' = 97\). This transition made it suitable to apply the stars and bars theorem, a combinatorial method that facilitates finding these solutions.

• Non-negative solutions are significant in combinatorics owing to their flexibility in allocation while maintaining the total sum at a constant value.
• The process involved reformulating the original variables to facilitate this approach.

By converting to non-negative solutions, a wider range of possibilities opens up, making the computation simpler and more comprehensive.