Question

Afour digit number is formed with the digits \(1,3,4,5\) without repetition. Find the chance that the number is divisible by 5 : (a) \(\frac{3}{4}\) (b) \(\frac{1}{4}\) (c) \(\frac{9}{16}\) (d) \(\frac{1}{16}\)

Short answer

Answer: (b) \(\frac{1}{4}\)

Step-by-step solution

Count the total permutations

First, we need to find the total permutations of the four-digit number formed with the digits \(1, 3, 4, 5\). There are 4 choices for the first digit, 3 choices for the second digit, 2 choices for the third digit, and 1 choice for the last digit (since no repetition is allowed). So, the total permutations are \(4\times3\times2\times1 = 24\).

Count the permutations ending in 5 and divisible by 5

To be divisible by 5, the number must end in 5. So, we have only one choice for the last digit. Then, there are 3 choices for the first digit, 2 choices for the second digit, and 1 choice for the third digit. Therefore, the number of permutations that form a four-digit number divisible by 5 is \(3\times2\times1 = 6\).

Calculate the probability

Now, we need to find the probability of forming a four-digit number that is divisible by 5. To do this, we will divide the number of permutations that are divisible by 5 by the total number of permutations. This will give us: Probability = \(\frac{6}{24}\) = \(\frac{1}{4}\)

Choose the correct option

We have found the probability to be \(\frac{1}{4}\). Therefore, the correct answer is (b) \(\frac{1}{4}\).

Key concepts

Permutation

In mathematics, "permutation" is the act of arranging all the members of a set into some sequence or order. Permutations are essential in probability as they help us determine potential outcomes.
In the exercise described, the focus is on arranging the digits 1, 3, 4, and 5 into different four-digit numbers without any repetition of digits. Here's how it works:

• You have 4 digits, and you want to arrange them specifically to form numbers.
• For the first digit, you're free to choose from any of the 4 digits.
• Once you've placed a digit, you have 3 choices left for the next position.
• This process continues until all digits are placed.

Thus, the count of all permutations for a 4-digit number using the digits 1, 3, 4, and 5 is computed as follows:
The first position has 4 choices, the second has 3, then 2, and finally 1. So, the total number of permutations is calculated by multiplying these choices: \[4 \times 3 \times 2 \times 1 = 24\]
This concept is crucial for setting the framework for evaluating probabilities.

Divisibility

Divisibility is a key concept used to determine if a number is capable of being evenly divided by another number without leaving a remainder.
For this particular exercise, the goal is to figure out which of the permutations of the digits 1, 3, 4, and 5 are divisible by 5.
To be divisible by 5, a number must end with the digit 5. This principle is useful in quickly identifying numbers that fit this criteria without needing complex calculations.

• Given the digits 1, 3, 4, and 5, to have a number ending in 5, one must place the digit 5 in the last position.
• This limits options for arranging the other digits, but it assures the condition for divisibility by 5.

In this context, the choice of 5 as the last digit influences the number of valid permutations. Once the last digit is fixed as 5, only the remaining three digits can be permuted among the first three positions, calculated as follows:
\[3 \times 2 \times 1 = 6\].
This calculation helps measure how many permutations are divisible by 5 out of the total possible permutations.

Number Theory

Number theory is a branch of pure mathematics devoted to the study of integers and integer-valued functions. It is the foundation for dealing with concepts like permutations and divisibility in our problem example. By focusing on integer properties, it helps students understand core aspects of the problem.
In this exercise, using number theory, we explored concepts such as:

• Identifying integers that are divisible by a given number (in this case, 5).
• Calculating total permutations using factorials, which are products of all positive integers up to a certain number, neatly represented in formulas like \(n!\).

Number theory provides a structured way to tackle these problems by offering clear principles for working with integers. This ensures that every step of finding the probability remains straightforward and logical.
Ultimately, understanding number theory assists in dividing permutations accurately to discern outcomes satisfying specific numerical properties, such as divisibility rules, helping simplify this probability-based task.