Question

A four digit number is formed by using the digits \(1,2,5,6\) and 8 without repetition. What is the probability that it will be an even number? (1) \(\frac{3}{5}\) (2) \(\frac{2}{5}\) (3) \(\frac{1}{2}\) (4) \(\frac{3}{10}\)

Short answer

Answer: The probability of forming an even four-digit number using the digits 1, 2, 5, 6, and 8 without repetition is 3/5.

Step-by-step solution

Counting Total Combinations

Let's first find the total number of four-digit numbers that can be formed with the given digits (1, 2, 5, 6, 8) without repetition. There are 5 digits we can choose for the first position, 4 remaining for the second, 3 remaining for the third, and 2 remaining for the last position. Therefore, the total number of combinations is \(5\times4\times3\times2 = 120\).

Counting Even Combinations

For a number to be even, it must end in an even digit. Since we have 3 even digits out of 5 digits in our list (2, 6, 8), the last position of the four-digit number can be any one of these three even numbers. Once we choose an even number for the last position, we have 4 remaining digits to choose for the first position, 3 remaining for the second, and 2 remaining for the third position. Therefore, the number of even combinations is \(4\times3\times2\times3 = 72\).

Computing Probability

Now that we have the total number of combinations and the number of even combinations, we can compute the probability of forming an even number. The probability is given by the ratio of even combinations over the total combinations: \(P(\text{even}) = \frac{72}{120}\) To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 24: \(P(\text{even}) = \frac{72\div24}{120\div24} = \frac{3}{5}\) Therefore, the probability that the four-digit number will be an even number is \(\boxed{\frac{3}{5}}\). (Answer: (1))

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns. It plays a crucial role in probability theory because it helps determine the number of ways an event can occur. When we talk about combinatorics in this context, we mean understanding how we can choose digits to form numbers without repetition.
In our exercise, we used combinatorics to determine the total number of four-digit numbers that can be formed from a set of five unique digits. By following the arrangements of these digits, we found a total of 120 possible combinations. This is foundational for calculating probabilities, as it sets the stage for determining specific outcomes, such as how many of these combinations result in an even number.

Permutations

Permutations refer to different ways of arranging a set of objects, where the order matters. In our exercise, we dealt with permutations when arranging the given digits to form distinct four-digit numbers.
The sequence in which digits are placed is critical, because the arrangement '1234' is different from '4321', even though the same digits are used. We calculated the number of permutations without repetition for our problem by multiplying the number of choices for each digit position: 5 choices for the first, 4 for the second, 3 for the third, and 2 for the last. This resulted in 5 × 4 × 3 × 2 permutations, which is essential for understanding how varied outcomes are possible in permutations.

Even Numbers

An even number is an integer divisible by 2, meaning it must end in 0, 2, 4, 6, or 8 in our context of digit-based numbers. For a four-digit number to be even, its last digit must be one of these even digits.
In the problem, we had three even digits (2, 6, and 8) out of five. Thus, for a number to be even, any one of these digits must occupy the last position. This reasoning was used to calculate the total number of even numbers that could be formed. Once an even digit is fixed at the end, permutations of the remaining digits were considered for the preceding positions, giving us 72 even-numbered permutations.

Class 10 Mathematics

Class 10 Mathematics introduces students to a variety of fundamental concepts, including probability and combinatorics. These topics are essential to understanding more advanced math and real-life applications like statistics.
This problem provides a practical application of these topics, helping students learn how to use combinatoric principles to solve probability problems. Such problems help reinforce the concept of structuring logical steps to reach a solution and simplifying complex-looking problems into manageable parts, a skill set critical for success in higher mathematics. The focus on organizing digits into permutations to find probabilities offers students an engaging way to apply theoretical knowledge to practical scenarios.