Problem 26 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples, is (A) 1332 (B) 666 (C) 333 (D) None of these

Step-by-Step Solution

Answer

666 ways.

Step 1: Define the Problem

We need to distribute 35 apples among 3 boys with no restrictions on how many apples each boy can receive. This is a combinatorics problem involving distributing indistinguishable items (apples) into distinguishable groups (boys).

Step 2: Apply Stars and Bars Theorem

The Stars and Bars Theorem is used to find the number of ways to divide a set of indistinguishable objects into groups. Using the formula \( \binom{n+k-1}{k-1} \), where \(n\) is the number of apples and \(k\) is the number of boys, we can calculate the number of distributions. Here, \( n = 35 \) and \( k = 3 \).

Step 3: Calculate Using the Formula

Calculate the number of distributions using the formula: \[ \binom{35+3-1}{3-1} = \binom{37}{2} \].

Step 4: Calculate the Binomial Coefficient

Compute \( \binom{37}{2} \), which represents choosing 2 positions for dividers among 37 total items (35 apples and 2 dividers).\[ \binom{37}{2} = \frac{37 \times 36}{2 \times 1} = 666 \].

Step 5: Conclusion

There are 666 ways to distribute the 35 apples among 3 boys with no restrictions on the number of apples each can receive.

Key Concepts

Stars and Bars Theorem

The Stars and Bars Theorem is a commonly used method in combinatorics that helps to distribute indistinguishable items into distinguishable groups. This theorem is particularly useful when you need to distribute objects, like apples, among different people or containers where the sequence doesn't matter.To put it simply, you can imagine the apples (stars) being separated by dividers (bars), which indicate the transition between groups. If you have 35 apples and 3 boys, you have to place 2 dividers between them, creating three sections for each boy.

• The formula used is \( \binom{n+k-1}{k-1} \).
• Here's what each symbol means:
  • \( n \) is the number of apples.
  • \( k \) is the number of groups (boys).
  • The formula determines how many ways you can place \( k-1 \) dividers among \( n+k-1 \) total items.

This way, the Stars and Bars Theorem allows for calculating distributions efficiently without manually accounting for every possibility.

Binomial Coefficient

The Binomial Coefficient is a key mathematical concept used to determine the number of possible ways to choose a subset of items from a larger set.In the context of our problem, it helps to decide how many different ways you can arrange dividers (bars) in the sequence of apples (stars).In the formula \( \binom{37}{2} \), you have:

• "37" representing the total number of positions (35 apples + 2 dividers).
• "2" representing the dividers you need to choose positions for.

The computation is given by the formula:\[\binom{37}{2} = \frac{37 \times 36}{2 \times 1} = 666\]This means there are 666 ways to select 2 divider positions from 37, ensuring that each boy gets a certain number of apples. It's an integral part of combinatorial calculations, especially when dealing with homogeneous groups.

Indistinguishable Objects Distribution

The distribution of indistinguishable objects is a significant concept in combinatorics. When dealing with items like apples that are identical, it's not about the items themselves but rather how they can be grouped or allocated to different recipients. Here, with 35 identical apples and 3 distinguishable boys, we aim to explore how they can share these apples without concern for which specific apple goes to which boy.

• The indistinguishability implies that any two distributions where the same number of apples are given to each boy are counted as the same.
• This simplifies the complex problem of finding all possible distributions.

Using the Stars and Bars Theorem helps here because it accounts for the indistinguishability, providing an efficient method to determine all possible ways to make the allocation. This way, you focus more on the combinatorial structures rather than the individual items, simplifying the problem significantly.