Problem 100 - Algebra for JEE ADVANCED

Question

Among the \(8 !\) permutations of the digits \(1,2,3, \ldots, 8\), consider those arrangements which have the following property. If we take any five consecutive positions, the product of the digits in these positions is divisible by \(5 .\) The number of such arrangements is equal to a. \(7 !\) b. \(2 .(7 !)\) c. \({ }^{7} C_{4}\) d. none of these

Step-by-Step Solution

Answer

The number of such arrangements is 20160; hence, the correct option is d) none of these.

Step 1: Understand the Condition

To satisfy the condition, the product of any five consecutive digits chosen from a permutation must be divisible by 5. This implies that at least one of the five digits must be 5, as it's the only eligible numeral from our set \( \{1, 2, 3, \ldots, 8\} \) that can ensure divisibility by 5.

Step 2: Position the Number 5

Given that the total number of positions is 8, and the requirements apply to any 5 consecutive positions, the number 5 must occupy one of the positions 4 through 8 in the permutation so that any such group of 5 digits will include 5.

Step 3: Calculate Possible Arrangements

There are 4 viable positions for the number 5: positions 4, 5, 6, 7, or 8. For each chosen position, the remaining 7 digits can be arranged in any order. So the number of valid permutations is calculating by \(4 \times 7!\).

Step 4: Calculate the Factorial Value

Calculate \(7!\), which represents the total arrangements of the remaining digits excluding 5. This results in \(7! = 5040\), so the total number of arrangements is \(4 \times 5040 = 20160\).

Step 5: Match to Option

Compare the calculated value (20160) to the provided answer options: a) \(7!\), b) \(2 (7!)\), c) \(^{7}C_4\), and d) none of these. Since our calculation (20160) does not match any of the options exactly, the correct answer is d) none of these.

Key Concepts

Divisibility

Divisibility refers to the ability of one number, say "A", to be divided by another number "B" without leaving a remainder. In simpler terms, when a number is divisible by another, it fits "perfectly" with no leftovers. This concept is crucial in number theory and helps solve problems involving gcd, lcm, and more. Divisibility rules make it easier to decide if a number can be divided by another, such as:

• A number is divisible by 5 if it ends in 0 or 5.
• For 2, the number must be even (last digit is 0, 2, 4, 6, or 8).
• A number divisible by 3 has digits that sum up to a multiple of 3.

In the exercise, the requirement was for the product of any five consecutive digits to be divisible by 5. As only the digit '5' itself can fulfill this condition within the numbers 1 through 8, it had to be strategically placed, ensuring any sequence of five included it. This concept makes divisibility not just simple arithmetics but also a logical puzzle, where placement and position determine the outcome.

Factorial

Factorials are an important concept in math, represented by the symbol "!" and denoting the product of an integer and all integers below it, down to 1. If you have a number like 8, expressed as 8!, it means:\[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\]Factorials are fundamental in permutations and combinations because they help provide the total number of ways objects can be arranged. They grow rapidly with even small increases in the base integer, making calculations both intriguing and challenging. In this problem, the factorial was used to derive the potential arrangements of numbers excluding the fixed number '5', calculated as 7!, to simplify solving for valid permutations.

Combinatorics

Combinatorics is the field of mathematics concerned with counting, both as a means and an end in obtaining results, and with exploring special structures and configurations that arise. This branch covers permutations, combinations, and beyond, allowing us to calculate the number of ways a set of objects can be arranged. In permutations, order matters. This was crucial in the problem at hand since every distinct arrangement of the digits mattered.

• Permutations consider exact positioning, like in our exercise, where specific placement secured divisibility.
• The formula for permutations of n items is simply n!.

In our exercise, even though only certain combinations of positions were viable for fulfilling the condition of product divisibility by 5, combinatorics helped determine how these digits could be arranged around the strategic placement requirement of the '5'. This is a classic example of how combinatorics turns abstract calculations into tangible problem-solving strategies.