Problem 15 - Algebra for JEE ADVANCED

Question

If \(N\) denotes the number of ways of selecting \(r\) objects out of \(n\) distinct objects \((r \geq n)\) with unlimited repetition but with each object included at least once in selection, then \(N\) is equal to a. \(\quad{ }^{r-1} C_{r-n}\) b. \({ }^{r-1} C_{n}\) c. \({ }^{r-1} C_{n-1}\) d. none of these

Step-by-Step Solution

Answer

The correct answer is option c: \(^{r-1}C_{n-1}\).

Step 1: Understanding the Problem

We need to determine the number of ways to select \(r\) objects from \(n\) objects when each object must be selected at least once, and repetition is allowed.

Step 2: Applying Combinatorial Logic

Since each object must be selected at least once, initially select one of each object. This leaves us with \(r-n\) selections to distribute freely among the \(n\) objects, with repetitions allowed.

Step 3: Reformulate Problem Using Stars and Bars

The problem of distributing \(r-n\) identical items into \(n\) distinct groups (with each group possibly empty) can be formulated as a "stars and bars" problem. The number of solutions is given by the formula \(^{x+k-1}C_{k-1}\), where \(x\) is the number of stars and \(k\) is the number of bars.

Step 4: Compute the Combination

In our reformulated problem, we have \(x = r-n\) stars and \(k = n\) bars (since there are \(n\) types of objects being chosen from). The number of combinations is \(^{r-n+n-1}C_{n-1} = ^{r-1}C_{n-1}\).

Step 5: Compare With Options

The expression obtained from our calculations, \(^{r-1}C_{n-1}\), matches option (c) from the available choices.

Key Concepts

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, a branch of mathematics that deals with counting and arranging objects. **Permutations** refer to the different ways to arrange a set of objects in a particular order. For example, if we have three books, labeled A, B, and C, the permutations of these books are ABC, ACB, BAC, BCA, CAB, and CBA. The order here is important.
**Combinations**, on the other hand, are selections of objects where the order doesn’t matter. If you are selecting two books from A, B, and C, the combinations are AB, AC, and BC. Here, the order AB is considered the same as BA.
This exercise focuses on combinations, particularly when selecting objects with repetition and under specific conditions. This requires understanding advanced techniques beyond basic combination formulas.

Stars and Bars Method

The stars and bars method is a powerful combinatorial tool used to solve problems involving distributing identical items into distinct groups. This method is especially useful when repetition is allowed, as seen in this problem. It works by imagining the items as 'stars' and the dividers as 'bars' between them.
For example, if you have to distribute 5 identical candies into 3 distinct bags, you visualize it as stars representing candies and bars representing the divisions between the bags. Pictorially, this could be represented for one possible solution as **\( \star \star | \star \star | \star \)**, meaning two candies in the first bag, two in the second, and one in the third.
In our exercise, after selecting one of each object, the problem reduces to finding the number of ways to distribute the remaining objects. This uses stars and bars with \(r-n\) stars and \(n-1\) bars, calculated via combinations.

Combination Formula

The combination formula, denoted as \( \binom{n}{r} \), is used to determine how many ways you can choose \(r\) objects from a set of \(n\) objects without regard to the order. The formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \(!\) denotes the factorial, meaning the product of all positive integers up to that number.
In scenarios involving repetition or constraints, such as each object being chosen at least once, the standard combination formula needs to be adapted. Our problem reformulated with stars and bars ends with a combination calculation \( \binom{r-1}{n-1} \), accounting for choosing slots for bars with given conditions.
This derived formula helps efficiently solve complex selection problems.

Discrete Mathematics

Discrete mathematics is a field concerned with distinct and separate values, often integers. It includes various topics such as graph theory, logic, and combinatorics, which tackles counting and arranging distinct objects—a key theme in this exercise.
Studying discrete mathematics allows understanding of algorithms, sequences, and structures that can be counted, arranged, or analyzed in finite steps.
This field is exceptionally relevant in computer science for data structure design, algorithm analysis, and handling subsets and permutations, much like the problems tackled in this exercise.
Combinatorics, the focus here, is a major branch of discrete mathematics and provides tools to handle problems involving selection, arrangement, and distribution, integrating logical reasoning to derive meaningful insights.