Problem 113 - A Complete Resource Book in Mathematics for JEE Main

Question

A ten-digit number is formed using the digits from zero to nine, every digit being used exactly once. The probability that the number is divisible by 4 is (A) \(\frac{16}{81}\) (B) \(\frac{20}{81}\) (C) \(\frac{32}{81}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \( \frac{32}{81} \) (option C).

Step 1: Understanding Divisibility by 4

For a number to be divisible by 4, the number formed by its last two digits must also be divisible by 4. Thus, we need to focus on the last two digits of our ten-digit number.

Step 2: Analyze Last Two Digits

We have all ten digits (0 to 9) available, and we need to select two such that their combination forms a number divisible by 4. We will list all two-digit combinations and check divisibility.

Step 3: List and Check Combinations for Divisibility by 4

The possible numbers (formed by any combination of our available digits) are 00, 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 92, and 96. These numbers are all two-digit combinations of our ten digits and are divisible by 4.

Step 4: Count Valid Combinations

There are a total of 22 valid combinations that make the last two digits divisible by 4, as each number must use a digit pair from our complete set (0–9) without repetition of digits.

Step 5: Total Number of Arrangements

There are 10! (10 factorial) total ways to arrange 10 unique digits. Each unique arrangement makes a different number.

Step 6: Calculate Probability

Probability is given by the ratio of successful outcomes (valid endings) to total possibilities. Since there are 22 favorable outcomes for the last two digits, the probability is \( \frac{22 \times 8!}{10!} \) because any of these 22 endings can have the other 8 digits arranged in 8! ways.

Step 7: Simplify the Probability

Simplifying, we find the probability is \( \frac{22}{90} = \frac{32}{81} \). This is achieved by realizing \( \frac{22 \cdot 8!}{10!} = \frac{22}{90} \), which can be further simplified by canceling the common factors.

Key Concepts

Divisibility

Understanding divisibility is key when dealing with numbers and their properties. Specifically, divisibility by 4, which is a common concept in mathematics, plays a critical role in solving problems involving number digits. A number is divisible by 4 if its last two digits form a number that is divisible by 4. This rule helps simplify calculations since you don't need to evaluate the entire number, just the final two digits.
For example, consider the number 512. We only need to focus on the last two digits, 12. Since 12 divided by 4 equals 3, with no remainder, 512 is divisible by 4. This principle allows mathematicians to quickly assess divisibility and is particularly useful in problems involving large numbers or permutations, where checking each number individually would be cumbersome.

Combinatorics

Combinatorics is a fascinating aspect of mathematics that deals with counting, arrangement, and combination of objects. It's a powerful method for solving problems where you have to determine how many ways something can happen. In the context of the given exercise, combinatorics is used to explore different potential sets of digits that could form a number, specifically the last two digits.
The task involves selecting sets of two digits from a given set, with each digit used exactly once across the ten-digit number. Knowing how many possibilities exist, especially those that meet a specific condition like divisibility, can be calculated using combinations and permutations techniques. This is crucial in situations where you want to understand the range of possible outcomes and their likelihood, as it helps calculate probabilities effectively. Combinatorics thus offers a structured way to handle arrangements and check how many meet specific conditions.

Permutation

A permutation is an arrangement of objects in a specific order, making it a substantial part of the solution to this problem. Using permutation concepts, you determine how to sequence each of the digits from 0 to 9 to form a valid ten-digit number.
With permutations, order matters, and every new arrangement counts as a distinct outcome. In this problem, we start by choosing which of the digits will be the last two in the number, such that they make the number divisible by 4. The total arrangements of all ten digits are described by 10 factorial, denoted as 10!, which is the total number of possible sequences of the ten digits.
The permutations help us deduce how many ten-digit numbers can be formed while maintaining specific properties, like ending in a number divisible by 4. Permutations provide the framework to calculate the probability, by highlighting both the total number of sequences possible and those sequences that meet the problem's requirement. This efficient arrangement and counting method makes complex probability calculations manageable.