Problem 11 - Algebra for JEE ADVANCED

Question

Number of ways of selecting three integers from \(\\{1,2,3, \ldots, n\\}\) if their sum is divisible by 3 is a. \(3\left({ }^{n / 3} C_{3}\right)+(n / 3)^{3}\) if \(n=3 k, k \in N\) b. \(2\left({ }^{\left(1 n-10^{\circ}\right.} C_{3}\right)+\left({ }^{(n+2) 3} C_{3}\right)+((n-1) / 3)^{2}(n+2)\), if \(n=3 k+1\), \(k \in N\) c. \(2\left({ }^{(n-1) \times 3} C_{3}\right)+\left({ }^{(n+2) 3} C_{3}\right)+((n-1) / 3)^{2}(n+2)\), if \(n=3 k+2\), \(k \in N\) d. independent of \(n\)

Step-by-Step Solution

Answer

The correct answer varies by case (a, b, or c) based on the form of \(n\). Option d is incorrect.

Step 1: Understanding the Question

We need to find the number of ways to select three integers from the set \( \{1, 2, 3, \dots, n\} \) such that their sum is divisible by 3. The question provides options depending on the form of \(n\): \(n = 3k\), \(n = 3k+1\), and \(n = 3k+2\), where \(k\) is a natural number.

Step 2: Analyze Case 1

For \(n = 3k\), the formula given is \(3\left({}^{n/3}C_3\right)+(n/3)^3\). This formula involves choosing groups from sets divisible evenly by 3, aiming for the sum of selected integers to also be divisible by 3.

Step 3: Analyze Case 2

For \(n = 3k+1\), it uses the formula \(2\left({}^{n-10^{\circ}}C_3\right)+{}^{(n+2)/3}C_3+((n-1)/3)^2(n+2)\). This case accounts for the remainder \(1\) when \(n\) is divided by 3 and adjusts accordingly.

Step 4: Analyze Case 3

For \(n = 3k+2\), the formula is \(2\left({}^{(n-1) \times 3}C_3\right)+{}^{(n+2)/3}C_3+((n-1)/3)^2(n+2)\). Again, adjustments are made for the case where \(n\) leaves a remainder \(2\) when divided by 3.

Step 5: Choose the Correct Option

The question gives specific scenarios for \(n = 3k\), \(n = 3k+1\), and \(n = 3k+2\) and relates these to possible solutions. None of these solutions describe a scenario where the solution is independent of \(n\). Thus, "d. independent of \(n\)" cannot be correct.

Key Concepts

Number Theory

Number theory is a field of mathematics focused on the properties and relationships of numbers, primarily integers. At its heart, it explores questions surrounding divisibility, the distribution of primes, and integer solutions to equations. It delves into understanding numbers through their structures and properties:

• Divisibility: This concept questions how some numbers can be wholly divided by others without any remainder. For example, 6 is divisible by 3, as 6 ÷ 3 equals 2, a whole number.
• Primes: Objects of great interest are prime numbers—integers greater than 1 with no positive divisors other than 1 and themselves.
• Congruences: A system where one studies the remainders of integer division. Problems like determining if a number is divisible by another often use this concept.

In combinatorics, number theory intersects to solve problems related to arrangements and selections, often using properties of integers. Understanding number theory can unravel complex combinatorial problems like the division and selection of integers from a set, as seen in integer partitions and divisibility rules.

Integer Partitions

Integer partitions involve breaking down a number into sums of integers. This concept is vital in various fields of mathematics, including combinatorics. In simple terms, an integer partition of a number is a way of writing it as a sum of positive integers where the order does not matter. Imagine trying to partition the number 5. The integer partitions would be:

• 5 = 5
• 5 = 4 + 1
• 5 = 3 + 2
• 5 = 3 + 1 + 1
• 5 = 2 + 2 + 1
• 5 = 2 + 1 + 1 + 1
• 5 = 1 + 1 + 1 + 1 + 1

This field of study helps to solve problems where a certain structure or counting method is needed around these partitions, especially in selecting combinations of numbers with properties like a sum divisible by a specific integer. It becomes especially useful when dealing with sums that have to satisfy particular divisibility conditions, such as those encountered in combinatorial number selections.

Divisibility Rules

Divisibility rules are shortcut methods that help determine if one number is divisible by another without performing the actual division. These rules are handy for quickly assessing divisibility and understanding how numbers behave relative to factors. Here are some essential rules:

• Rule for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 123 (1 + 2 + 3 = 6) is divisible by 3.
• Rule for 5: A number is divisible by 5 if its last digit is 0 or 5.

These rules align with the core aspects of the exercise, which involves determining combinations of numbers where their total sums are divisible by a specified number. Employing these rules helps simplify the understanding and calculations of such problems, especially when organizing numbers into divisible groups. For instance, ensuring that the sum of selected integers is divisible by a number like 3 can be quickly verified using these rules before moving on to further calculations.