Problem 109 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of ways in which 16 identical things can be distributed among 4 persons if each person gets at least 3 things, is (A) 33 (B) 35 (C) 38 (D) None of these

Step-by-Step Solution

Answer

35 ways.

Step 1: Distribute Minimum Things

Each person must receive at least 3 things. Since we have 4 people, we distribute \(3 \times 4 = 12\) things initially. This leaves us with \(16 - 12 = 4\) things to be distributed freely among the 4 people.

Step 2: Distribute Remaining Things Using Stars and Bars

We now have 4 identical things left to distribute among 4 persons. This is a combinatorial "stars and bars" problem where we find the number of non-negative integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 4\). The formula for this is \(\binom{n + k - 1}{k - 1}\), where \(n\) is the number of things to distribute and \(k\) is the number of people.

Step 3: Apply Stars and Bars Formula

In this case, \(n = 4\) and \(k = 4\). Therefore, the number of solutions is \(\binom{4 + 4 - 1}{4 - 1} = \binom{7}{3}\).

Step 4: Calculate the Combination

Calculate \(\binom{7}{3}\) which is \(\frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\). Hence, there are 35 ways to distribute the 4 remaining things.

Key Concepts

Stars and Bars Method

The "Stars and Bars" method is a powerful tool in combinatorics. It is used to find the number of ways to distribute identical items into distinct groups. Imagine the stars as the items to distribute, and bars as dividers between different groups.
This technique is particularly useful for solving problems where you're asked to determine the number of positive or non-negative integer solutions to an equation.

• Stars: Represent the identical items.
• Bars: Represent the separations between groups.

Using the "Stars and Bars" method, the number of solutions to distributing \(n\) indistinguishable items into \(k\) distinguishable boxes (where boxes can hold any number of items) is given by the formula: \[\binom{n + k - 1}{k - 1}\] This formula arises from the concept that you need \(k-1\) bars to create \(k\) groups. Together with \(n\) stars, you have a total of \(n + k - 1\) places to choose from to place the bars, hence the combination formula.

Combinatorial Problems

Combinatorial problems involve counting, arranging, and selecting without actually listing all the options. They are key in fields like mathematics and computer science.
Combinatorics helps answer questions like "How many ways can we distribute items?" without exhaustive enumeration. These problems often use specific types of mathematical tools, like permutations and combinations.

• Counting: Directly counting the number of ways to achieve a set-up (like items in a group).
• Permutations: Arrangements where order matters.
• Combinations: Groups where order doesn’t matter.

The challenge lies in matching a scenario to the correct mathematical tool to solve it efficiently. For instance, distributing 16 identical items so each of 4 people receives at least 3 is solved using combinations because the order doesn’t matter, just the way the items are grouped.

Integer Solutions

Finding integer solutions is common in many combinatorial problems. In the context of distributing items, it often boils down to solving an equation with non-negative solutions.Consider distributing \( n \) identical items into \( k \) distinct groups. Typically, the problem is expressed as finding non-negative integer solutions to an equation like \[x_1 + x_2 + \cdots + x_k = n\]where each \(x_i\) represents items in group \(i\).

• Non-negative solutions: Allow zero items in some groups.
• Positive solutions: Require each group to have at least one item.

Using the stars and bars method, you can find how many ways exist to satisfy these conditions efficiently, which is crucial especially when dealing with larger numbers. The ability to break down these equations into understandable parts makes solving such combinatorial problems easier and clearer.