Question

The digits \(1,2,3,4,5,6,7,8,9\) are written in random order to form a nine digit number. Find the probability that this number is divisible by 4 : (a) \(\frac{4}{9}\) (b) \(\frac{2}{9}\) (c) \(\frac{17}{81}\) (d) none of these

Short answer

(a) 1/9 (b) 1/3 (c) 1/4 (d) None of these Answer: (d) None of these.

Step-by-step solution

Identify the possible multiples of 4

Find all the possible pairs of two digits from the given digits, which give multiples of 4 when combined together. These pairs are: (2,4), (4,8), (3,2), (6,4), (8,12), (1,6), (6,8), (4,4), (7,6), (7,2), (8,4), and (8,8).

Calculate the total number of ways to arrange nine digits

There are 9 digits in the total digits, and each digit can be placed in any of the positions. So, there are 9! ways to arrange these digits to form a nine-digit number.

Calculate the number of favorable ways for each pair of digits (those which form multiples of 4)

We will count the number of ways of arranging the digits such that the selected pair of digits (that formed a multiple of 4) appears in the last two positions. (a) For (2,4): The pair (2,4) can be placed in 2! ways and the remaining 7 digits can be placed in 7! ways. So, the number of favorable ways for (2,4) is 2! * 7!. (b) Similarly, for (4,8): 2! * 7!. (c) For (3,2): 2! * 7!. (d) For (6,4): 2! * 7!. (e) For (8,12): 2! * 7!. (f) For (1,6): 2! * 7!. (g) For (6,8): 2! * 7!. (h) For (4,4): In this case, the two 4s are indistinguishable, so there is only 1 way to arrange them. The remaining 7 digits can be arranged in 7! ways. Therefore, the number of favorable ways for (4,4) is 7!. (i) For (7,6): 2! * 7!. (j) For (7,2): 2! * 7!. (k) For (8,4): 2! * 7!. (l) For (8,8): Similarly to the case with (4,4), there is only 1 way to arrange the two 8s, and the remaining 7 digits can be arranged in 7! ways. Thus, the number of favorable ways for (8,8) is 7!.

Calculate the total number of favorable ways

Add the favorable ways for each of the pairs: 10 * (2! * 7!) + 2 * (7!) = 2 * 7! * (10 + 1) = 22 * 7!.

Calculate the probability

Divide the total number of favorable ways by the total number of ways to arrange the digits: Probability = \(\frac{22 * 7!}{9!} = \frac{22}{9*8} = \frac{11}{36}\). Since this fraction is not included in options (a), (b), and (c), the correct answer is (d) None of these.

Key concepts

Divisibility

Divisibility is the property that determines if one integer can be divided by another without leaving a remainder. For a number to be divisible by another, after division, the remainder must be zero. In the exercise, we are interested in numbers divisible by 4. A quick test for divisibility by 4 is to check the last two digits of the number. If these two digits form a number that is divisible by 4, then the entire number is divisible by 4.

Understanding divisibility rules helps simplify calculations and solve problems involving large numbers, like determining how many permutations of a set are divisible by a specific number. In this way, knowing and applying divisibility rules can be a great time-saver in problems like the one provided in the exercise.

Permutation

Permutation refers to the arrangement of a set of objects in a specific order. For any given set of "n" objects, the number of permutations can be calculated as the factorial of "n", or "n!". In the exercise, we are asked to consider permutations of nine different digits to form a number.

The total number of permutations of nine digits, like in the problem, is calculated as:

• 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

This calculation helps us understand the total possible ways we can arrange those digits to form each distinct number. Understanding permutations is crucial in problems involving order and arrangement, like forming numbers or in arranging items in various combinations.

Factorial

A factorial, denoted by an exclamation point "n!", is the product of all positive integers up to a specific number "n". It calculates the number of ways to arrange "n" objects.

For example, "5!" means multiplying all integers from 1 to 5:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

In the exercise, factorials are used to determine how many different ways digits can be arranged. For nine digits, this is calculated as "9!". Factorial calculations are foundational in combinatorics and the study of permutations and combinations, helping determine outcomes in complex arrangement scenarios.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. It's the art of counting things and arranging them in systems. Each problem will have specific rules related to counting, like how many ways you can distribute, organize, or select items.

In this problem, combinatorics is at play when assessing the probability of forming a number divisible by 4. We examine different pairs of the last two digits to see how they can be arranged to meet the divisibility rule.

• We create sets of favorable and total outcomes.
• We calculate the probability as the ratio of favorable outcomes to total outcomes.

Through combinatorics, we can accurately solve problems by systematically counting possible outcomes and making solutions more manageable.