Problem 162 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \({ }^{9} C_{3}\) [2011] Statement- \(\mathbf{2}\) : The number of ways of choosing any 3 places from 9 different places is \({ }^{9} C_{3}\). (A) Statement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement- 1 (B) Statement-1 is true, Statement-2 is false. (C) Statement- 1 is false, Statement- 2 is true. (D) Statement- 1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

Step-by-Step Solution

Answer

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Step 1: Understanding Statement 1

Statement 1 is about distributing 10 identical balls into 4 distinct boxes such that no box is empty. Using the "stars and bars" theorem, if each box must contain at least one ball, we first place one ball in each box (4 balls), leaving us with 6 balls. We then need to partition 6 identical balls into 4 partitions (boxes), which is equivalent to finding the non-negative integer solutions of the equation: \( x_1 + x_2 + x_3 + x_4 = 6 \). This is solved using combinations: \( \binom{6+3}{3} = \binom{9}{3} \). So, Statement 1 is true: \( ^9C_3 \).

Step 2: Understanding Statement 2

Statement 2 refers to choosing 3 places from 9 different ones, which is a straightforward calculation of combinations: \( \binom{9}{3} \). This setup indeed calculates how many ways we can select any 3 locations from a total of 9, which gives us \( ^9C_3 \). Thus, Statement 2 is true as well.

Step 3: Analyzing the Relationship Between Statements

Both statements yield the same combination expression, \( ^9C_3 \), but they originate from different contexts. Statement 1 uses the expression as part of a distribution problem using "stars and bars," while Statement 2 simply involves selecting distinct items. Therefore, while both statements are true, Statement 2 does not explain Statement 1.

Key Concepts

Stars and Bars Method

The Stars and Bars Method is a combinatorial technique used to solve problems involving distributing indistinguishable objects (like identical balls) into distinguishable boxes (like distinct containers). Imagine you have 10 identical balls and 4 distinct boxes and you want every box to have at least one ball. Here's how it works:

* First, put one ball in each box to ensure no box is empty. This leaves us with 6 balls to distribute.
* Now, imagine the problem as finding non-negative solutions to the equation: \( x_1 + x_2 + x_3 + x_4 = 6 \).

To solve it, we use the concept of separating the remaining 6 balls into 4 groups using 3 bars. The balls are stars, and the bars divide them, hence the name "stars and bars." This is equivalent to choosing positions for the 3 bars among the 9 total places (6 stars + 3 bars) which gives us \( \binom{9}{3} \) ways.

The stars and bars method simplifies complex-looking distribution problems and turns them into manageable combination calculations.

Combination Formula

The combination formula is a fundamental tool in combinatorics. It lets us calculate the number of ways to choose a subset of items from a larger set, without considering the order. When you're choosing "k" items from a total of "n" items, the combination formula applies:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

In our context, such as deciding where to place the bars in the stars and bars method, this formula becomes crucial. For instance, if you're selecting 3 places for the bars out of a total of 9 (6 stars + 3 bars), you calculate it as \( \binom{9}{3} \). This is the same formula used when choosing 3 locations from 9 in a straightforward selection problem from Statement 2 in the original exercise.

By using the combination formula, complex distribution and selection problems become manageable with precise calculations, helping you quickly discover the number of possible outcomes.

Non-negative Integer Solutions

Finding non-negative integer solutions is common in distribution problems. These solutions help determine how many ways there are to distribute items such that no quantity is negative. For example, if you want to distribute 6 balls into 4 boxes, allowing some boxes potentially to be empty, solve for non-negative integers in the equation \( x_1 + x_2 + x_3 + x_4 = 6 \).

In exercises like Statement 1, where each box gets at least one item, you first assign one item to each box, then solve for the leftover items with non-negative solutions. This technique is integral to the stars and bars method, and it ensures every "box" gets a fair share of the "stars."

Understanding how non-negative integer solutions work is essential in tackling problems involving partitioning and distribution. It helps in framing problems with constraints in a way that the combination formula can solve effectively, ensuring that proper counts of possible distributions are reached without requiring intricate individual calculations.