Problem 13 - Algebra for JEE ADVANCED

Question

Number of shortest ways in which we can reach from the point \((0,0,0)\) to point \((3,7,11)\) in space where the movement is possible only along the \(x\)-axis, \(y\)-axis and z-axis or parallel to them and change of axes is permitted only at integral points (an integral point is one which has its coordinate as integer) is a. equivalent to number of ways of dividing 21 different objects in three groups of size \(3,7,11\) b. equivalent to coefficient of \(y^{3} z^{7}\) in the expansion of \((1+y+z)^{21}\) c. equivalent to number of ways of distributing 21 different objects in three boxes of size \(3,7,11\) d. equivalent to number of ways of arranging 21 objects of which 3 are alike of one kind, 7 are alike of second type and 11 are alike of third type

Step-by-Step Solution

Answer

The answer is option d.

Step 1: Understanding the Problem

We need to find the number of shortest paths from point \((0,0,0)\) to point \((3,7,11)\), where travel is restricted to moving along or parallel to the axes, and can only occur at integer points. Each path corresponds to a sequence of 21 moves: 3 moves along the x-axis, 7 along the y-axis, and 11 along the z-axis.

Step 2: Total Moves Required

The total number of moves required is the sum of moves along each axis: 3 moves on the x-axis, 7 on the y-axis, and 11 on the z-axis. Hence, the total number of moves is \(3 + 7 + 11 = 21\).

Step 3: Define Unique Moves Configuration

As each axis move is distinct and can be thought of as moving a unique 'object' per axis, the unique configuration for arranging these movements can be compared to arranging 21 items where 3 items are of one kind, 7 of another, and 11 of another.

Step 4: Use of Combinatorial Formula

The number of ways to arrange these moves is given by the formula for permutations of multiset: \(\frac{21!}{3!7!11!}\), where the factorials in the denominator account for indistinguishable moves along the same axis.

Step 5: Identify Equivalent Provided Options

Among the given options, option (d) describes dividing 21 objects such that 3 are alike of one kind, 7 of another kind, and 11 of yet another kind, which matches the mathematical description of the required moves along the axes perfectly.

Step 6: Calculate the Answer

Using the formula: \(\frac{21!}{3!7!11!}\), calculate to confirm equivalence with given options.

Key Concepts

Combinatorial Paths

Combinatorial paths provide a way to understand the number of distinct routes one can take when moving from one point to another in a structured manner. In the problem, we are tasked with moving from the starting point \((0,0,0)\) to the point \((3,7,11)\) using the shortest possible path. Here, the 'shortest path' doesn't mean the literal shortest distance. Instead, it refers to the path using the minimum number of moves allowed by the constraints, moving exactly along defined axes.
Think of it like having a set number of steps in each direction:

• 3 steps in the x-axis,
• 7 steps in the y-axis, and
• 11 steps in the z-axis.

The challenge is to organize or sequence these moves so we reach the final destination point. This type of problem is a classic example of how combinatorics is used to solve path-finding in structured environments. It's not about taking shortcuts diagonally or in unconventional ways; each axis's steps must be completed in the prescribed manner.
Using combinatorial paths, we can determine the number of ways to execute the required steps efficiently.

Multiset Permutations

Multiset permutations arise when we need to arrange a collection of items where some of the items are indistinguishable from each other. Consider the problem of moving along axes: we have 21 moves that need arranging, but moves along the same axis are indistinguishable, like indistinguishable 'objects'.

To calculate the number of distinct permutations, we use the multiset permutation formula:\[\frac{21!}{3!7!11!}\]
This formula computes the number of ways to arrange our moves (or objects) where:

• Each axis represents a different 'group' of indistinguishable moves,
• The factorial in the denominator accounts for these indistinguishable nature of moves along particular axes.

Because the paths involve repeating types of moves (like repeated steps along the same axis), we apply this formula to identify how many distinct sequences or paths can be created.
Understanding multiset permutations helps in solving complex arrangement problems like ours, where not every component is unique but can still result in complex ordered outcomes.

3D Coordinate System

Using a 3D coordinate system allows us to navigate and describe points in space using three values representing the x, y, and z positions. In this system, each integral point has integer-valued coordinates, which means each axis's movement is tracked in whole integer steps.
In our exercise, we start at point \((0,0,0)\), a basic reference point, and aim to reach \((3,7,11)\), our target point, by moving axis-aligned paths. The 3D coordinate system ensures that our navigation is systematic, with each axis contributing its part to the overall shift in position:

• The x-axis adjustment by 3 units,
• The y-axis adjustment by 7 units,
• The z-axis adjustment by 11 units.

The beauty of the 3D system lies in its flexible structure, accommodating not only straightforward directional movements but also enabling complex path computations through combinatorial methods.
Understanding coordinate systems enriches problem-solving, especially in spatial scenarios like constructing paths or predicting trajectory shifts. It provides a reliable framework to visualize changes and compute paths effectively.