Problem 135 - A Complete Resource Book in Mathematics for JEE Main

Question

There are five different boxes and seven different balls. All the seven balls are to be distributed in the five boxes placed in a row so that any box can receive any number of balls. Suppose, all the balls are identical, then in how many ways can all these balls be distributed into these boxes? (A) 110 (B) 220 (C) 330 (D) 1440

Step-by-Step Solution

Answer

The balls can be distributed in 330 ways.

Step 1: Understand the Problem

We have 7 identical balls to distribute across 5 distinct boxes. We're looking for the number of ways to distribute these balls where any box can hold 0 to 7 balls.

Step 2: Set Up the Equation

We will use the stars and bars method for this combinatorial problem. The formula for distributing \( n \) identical balls into \( k \) distinct boxes is given by \( \binom{n+k-1}{k-1} \). In our problem, \( n = 7 \) (balls) and \( k = 5 \) (boxes).

Step 3: Plug Values into the Formula

Substitute \( n = 7 \) and \( k = 5 \) into the formula: \[ \binom{7+5-1}{5-1} = \binom{11}{4} \]

Step 4: Calculate the Combination

Now calculate \( \binom{11}{4} \), which is the number of ways to choose 4 objects from a set of 11. This can be calculated as: \[ \binom{11}{4} = \frac{11!}{4! \times (11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330 \]

Step 5: Interpret the Result

The calculation shows there are 330 ways to distribute the 7 identical balls into the 5 boxes.

Key Concepts

Stars and Bars method

The Stars and Bars method is a crucial concept in combinatorics for solving problems involving the distribution of identical objects into distinct groups. It is particularly useful when determining the number of ways to assign items where order does not matter. In this method, the objects to be distributed are represented as 'stars', and the dividers between different groups are represented as 'bars'.

Consider the problem of distributing 7 identical balls into 5 distinct boxes. You can visualize this as placing stars in a row, separated by bars to indicate group boundaries. The goal is to insert enough bars (or dividers) amongst the stars to form the required groups.

• Stars represent the items being distributed (7 balls).
• Bars represent the divisions needed to separate groups (4 bars for 5 boxes).

Thus, the problem becomes how to choose positions for 4 bars in a total of 11 slots (7 stars + 4 bars), which leads us to use the combination formula.

Combination formula

The combination formula is essential when you need to calculate the number of ways to choose a subset of items from a larger set without regard to order. In mathematics, this is represented as \( \binom{n}{r} \), which is read "n choose r". The formula is calculated as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Where:

• \( n! \) is the factorial of the total number of items.
• \( r! \) is the factorial of the number we are choosing.
• \((n-r)!\) is the factorial of the difference between the total and the chosen number.

In the context of the Stars and Bars method outlined in the exercise, we use the combination formula to determine how to select the placement of the "bars" within the sequence of "stars and bars". For the given problem with 7 balls and 5 boxes, our calculation was \( \binom{11}{4} \), resulting in 330 ways.

Identical objects distribution

When solving combinatorial problems involving identical objects, one key aspect is to understand that the arrangement doesn't depend on the distinctness of objects. Each object is indistinguishable from the others, making it different from problems where you distribute distinct items.

In our exercise, we're distributing 7 identical balls into 5 distinct boxes. Since the balls are identical, the primary concern is the number allocated to each box and not the specific arrangement of any particular ball.

This approach reduces complexity because we only consider the number of balls per box, not the uniqueness of each ball. This can often make calculations simpler, as it turns a potentially complicated permutation problem into a straightforward application of the stars and bars technique. With this understanding, the combination approach gives us the required number of distributions, ensuring that each distribution is accounted for without overcounting due to object indistinguishability.