Problem 2 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is : \(\quad\) [Jan. 9,2020 (II)] (a) \(\frac{965}{2^{11}}\) (b) \(\frac{965}{2^{10}}\) (c) \(\frac{945}{2^{10}}\) (d) \(\frac{945}{2^{11}}\)

Step-by-Step Solution

Answer

The probability is \(\frac{945}{2^{11}}\).

Step 1: Calculate Total Arrangements

First, calculate the total ways to distribute 10 different balls into 4 distinct boxes. Since each box can contain any ball, and all balls are different, the total number of arrangements is given by \(4^{10}\).

Step 2: Select Boxes for 2 and 3 Balls

Now, choose two boxes which will contain exactly 2 and 3 balls respectively. The number of ways to choose 2 boxes out of 4 is given by the combination formula \(\binom{4}{2} = 6\).

Step 3: Distribute Balls Among Selected Boxes

From the 10 balls, choose 2 to place in the first selected box. The number of ways to choose 2 balls from 10 is \(\binom{10}{2} = 45\). Choose 3 balls from the remaining 8 balls to place into the second box. The number of ways to choose 3 balls from 8 is \(\binom{8}{3} = 56\).

Step 4: Distribute Remaining Balls

Place the remaining 5 balls into the remaining 2 boxes. Each of these balls can be placed in either of the two remaining boxes. Thus, there are \(2^5 = 32\) ways to distribute these 5 balls.

Step 5: Calculate Favorable Arrangements

Multiply the results from steps 2, 3, and 4 to find the total number of favorable arrangements. Therefore, it is \(6 \times 45 \times 56 \times 32\).

Step 6: Calculate Probability

Finally, calculate the probability by dividing the number of favorable arrangements by the total number of arrangements: \[ P = \frac{6 \times 45 \times 56 \times 32}{4^{10}} = \frac{945}{2^{11}} \]. Therefore, the correct answer is (d).

Key Concepts

Combinatorial Probability

Combinatorial probability involves using combinatorial methods—methods that involve counting combinations and permutations—to determine the likelihood of complex, discrete outcomes. It's a key concept when dealing with situations where the arrangement of items figures prominently in the problem.
For example, when dealing with the problem of placing 10 different balls into 4 distinct boxes, we use combinatorial probability to determine the probability of two specific boxes each containing an exact number of balls. In this case, "10 different balls" emphasizes the need for arranged, non-repetitive allocation, meaning each ball ends up in its own unique configuration among the boxes selected.

The consideration of all possible distributions, through calculations such as selecting boxes and arranging remaining items, forms the basis of understanding probabilities in combinatorics. This involves dividing favorable outcomes by all possible outcomes to get the probability of a given event. Thus, mastering combinatorial probability provides essential tools for solving complex problems in discrete situations.

Discrete Mathematics

Discrete mathematics is a field of study that deals with distinct and separate values, as opposed to continuous mathematics, which deals with quantities that can vary continuously. This branch of mathematics includes topics such as logic, set theory, and, relevant to our exercise, combinatorics.
The problem of determining the probability that two boxes contain exactly 2 and 3 balls respectively from a set of 10 different balls is a typical example of discrete mathematics. Discrete mathematics does not require knowledge of calculus or other continuous topics, focusing instead on finite concepts.

In this exercise, discrete mathematics is employed to manage selections and placements, considering finite options like the number of boxes or balls. Calculations involve determining the number of combinations and permutations of these discrete items. Understanding discrete mathematics can thus empower students to solve problems involving distinct, countable structures more effectively.

Permutations and Combinations

Permutations and combinations are fundamental tools in combinatorics, a branch of mathematics dealing with counting, arranging, and ordering items. While permutations consider the order of items, combinations ignore order. Both are essential for solving problems where items are grouped, distributed, or arranged.
In the problem at hand, combinations are used to determine how different scenarios can be selected or occur. For example, combination formulas are utilized to select 2 boxes out of 4, select 2 balls from 10 for one box, and 3 balls from 8 for another box. This systematic selection ensures that all configurations are accounted for and aids in calculating the required probability.

Permutations could also come into play if the sequence in which the balls are placed was relevant. Thus, permutations and combinations provide the mathematical structure necessary for solving problems requiring specific object arrangements and selections.