Problem 12 - A Complete Resource Book in Mathematics for JEE Main

Question

A five digit number divisible by 3 is to be formed using the numerals \(0,1,2,3,4\) and 5 without repetition. The total number of ways this can be done is (A) 216 (B) 600 (C) 240 (D) 3125

Step-by-Step Solution

Answer

240 ways.

Step 1: Understanding Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. Here, we have the digits 0, 1, 2, 3, 4, and 5. The sum of all these digits is 15. Since 15 is divisible by 3, any combination of these digits that uses all six will also be divisible by 3.

Step 2: Formulating a Five-Digit Number

Since a five-digit number cannot start with 0, we will divide the task into two scenarios: one where 0 is not included in the number of five digits, and two further subcases where it is included but not at the start.

Step 3: Case 1: Numbers Without 0

For a five-digit number formed only using the digits 1, 2, 3, 4, and 5, which total 15 (and are divisible by 3), calculate the number of permutations. Since there are 5 distinct digits, there are 5! (120) possible permutations.

Step 4: Case 2: Numbers With 0

Include 0 and calculate based on this setup. Choose 4 out of the remaining 5 digits from 1, 2, 3, 4, and 5 to form the rest. Like case 1, these digits also sum to a number divisible by 3. Calculate the permutations but exclude those beginning with 0.

Step 5: Sub-case 2.1: Choose and Order

From digits {1, 2, 3, 4, 5}, choose 4 digits in \(\binom{5}{4}\) ways, i.e., 5 ways. Then include 0 and permute these 5 digits. For each choice, ignore permutations starting with 0. So, for each choice, valid permutations are 4!. This yields \(5 \times 24 = 120\) valid numbers.

Step 6: Add Total Permutations

Combine the valid format from both cases. Add 120 (without 0) to 120 (with 0 but not starting with it) to find the total number of ways to create the number.

Key Concepts

Divisibility Rules

Divisibility rules help in determining whether a number can be divided evenly by another number.
For divisibility by 3, the rule is straightforward: a number is divisible by 3 if the sum of its digits is divisible by 3. For instance, the number 123 is divisible by 3 because the sum of its digits, 1 + 2 + 3, is 6, which is divisible by 3.

In the provided exercise, we are working with the digits 0, 1, 2, 3, 4, and 5. The sum of these digits is 15. Since 15 is divisible by 3, any combination of these digits will have a sum that is divisible by 3. This foundational rule helps us quickly identify which combinations result in numbers divisible by 3 without needing further calculations. It simplifies our task of forming a number that meets this criterion.

Factorial

The concept of factorial, denoted as "!", is fundamental in permutations and combinations. Factorial of a number, say n, is the product of all positive integers less than or equal to n. For example, the factorial of 5, written as 5!, is 5 × 4 × 3 × 2 × 1 = 120.

Factorials are useful in determining the number of ways unique items can be arranged or permuted. They are crucial in combinatorics, where we calculate permutations or combinations of items.
In the exercise, factorials are used to determine how many unique five-digit numbers can be formed from a set of digits. When the task specifies using all available digits, factorial calculations let us easily compute the total number of possibilities.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations of sets. It answers questions like how we can select and arrange items from a larger set.

When working with permutations, the order of items matters. For a set of n items, the number of permutations of all n items is n!. When creating five-digit numbers from larger sets, like in our exercise, combinatorics helps us determine how many subsets of digits can be selected and the number of unique arrangements possible for each subset.
The solution involves choosing 4 digits from a set of 5, i.e., using the combination formula \( \binom{n}{r} \) where \( n \) is the total items, and \( r \) is the chosen items.

Number Formation

Number formation involves selecting digits to create numbers with specific attributes, like divisibility. When forming numbers, the initial digits have constraints, such as not starting with zero to maintain the required digit count.

In the exercise discussed, we are forming a five-digit number divisible by 3 from six available digits. We examine different cases:

• Forming numbers without including 0, ensuring the total remains divisible by 3.
• Including 0 but not allowing it as the leading digit, conforming to five-digit expectations.

This systematic approach ensures all potential combinations are considered, while adhering to the requirements and leveraging properties like divisibility.