Problem 19 - Algebra for JEE ADVANCED

Question

Total number of six-digit numbers in which all and only odd digits appear is a. \(\frac{5}{2}(6 !)\) b. \(6 !\) c. \(\frac{1}{2}(6 !)\) d. none of these

Step-by-Step Solution

Answer

The total number of such six-digit numbers is 1800, so the answer is d. none of these.

Step 1: Identify Odd Digits

First, identify the odd digits that can be used in a six-digit number. These are 1, 3, 5, 7, and 9, giving a total of 5 different odd digits.

Step 2: Calculate Total Possible Six-Digit Numbers

We need to form a six-digit number using only odd digits, and each digit must be used exactly once. Since there are 5 odd digits, we must have 1 repeated digit to make up 6 digits in total.

Step 3: Choose the Repeated Digit

Select one of the 5 odd digits to repeat. This means there are 5 ways to choose which digit will appear twice in the number.

Step 4: Arrange the Digits

Once a digit is chosen to be repeated, we need to arrange these 6 digits (where one is repeated). The formula for arranging n elements with repetition is \( \frac{n!}{k!} \), where \(k\) is the number of times the repeated element occurs. Here, the arrangement becomes \( \frac{6!}{2!} \).

Step 5: Calculate Total Combinations

Since there are 5 choices for the repeated digit and \( \frac{6!}{2!} \) ways to arrange the digits, the total number of distinct six-digit numbers is \( 5 \times \frac{6!}{2!} \).

Step 6: Simplify the Expression

Calculate \( \frac{6!}{2!} \): - \(6! = 720\), and \(2! = 2\), - So, \( \frac{720}{2} = 360\).Multiply by 5: - Total = \( 5 \times 360 = 1800\).

Key Concepts

Odd digits

Odd digits are the numbers that cannot be evenly divided by two. In simpler terms, they have a remainder of 1 when divided by 2.
For example, the sequence of odd digits includes numbers like 1, 3, 5, 7, and 9.
This means they are not part of the typical even numbers you might find in lists. Picking numbers to form a particular sum or to fit within certain criteria is common in permutations and combinations. In this exercise, these odd digits are used to form six-digit numbers. Every digit must be used, and one must repeat.
Understanding which digits are considered odd is important because it directly affects how you can form the numbers.

Factorial

The factorial of a number—designated by an exclamation point (e.g., 6!)—is the product of all positive integers up to that number.
Factorials are crucial in permutations because they determine how many ways you can arrange a set of items.For example:

• The factorial of 3 (3!) is calculated as 3 × 2 × 1 = 6.


• In the context of this exercise, calculating 6! (6 factorial) means multiplying
  1. 6 × 5 × 4 × 3 × 2 × 1 = 720

Factorials simplify expressions used to calculate permutations, especially when determining arrangements with repetition. Repetition of digits is managed using the formula \(\frac{n!}{k!}\), reflecting this relationship in the problem.

Six-digit numbers

Creating six-digit numbers involves selecting digits to fill each of the six positions.
In scenarios where restrictions apply, like only using odd digits, understanding permutations becomes necessary.In this task, a six-digit number is formed exclusively using the odd digits: 1, 3, 5, 7, and 9. With only five distinct odd digits available, one digit must repeat to create a six-digit sequence.
This decision gives rise to the concept of permutation with repetition.Here's how it works:

• Choose a digit to repeat. You have 5 choices here.


• Arrange the six digits including the repeated one. This is calculated using \(\frac{6!}{2!}\).


• Multiply these permutations by the number of choices for the repeated digit.

Finally, calculate these values for the total number of unique six-digit numbers possible.Thus, recognizing and properly applying these concepts allows you to solve permutation problems effectively.