JEE Main 2023 (24 January Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Permutations and Combinations: The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ____.

Enter Numerical Value:

Progressive Hints

Hint 1

Hint 1: Visualize the 9-digit number as a sequence of 9 slots. Identify which slots are 'even' (2nd, 4th, 6th, 8th) and which are 'odd' (1st, 3rd, 5th, 7th, 9th). The constraint is a strict gatekeeper: only even digits are allowed in the even slots.

Hint 2

Hint 2: Once you have separated the digits into two distinct pools—the even pool (2, 2, 4, 4) and the odd pool (1, 1, 1, 3, 3)—remember that the order within these pools matters, but you must account for the identical digits using the permutation formula for multisets:

\frac{n !}{n _{1} ! n _{2} ! \dots}

Hint 3

Hint 3: The total number of arrangements is the product of the independent ways to arrange the even digits in their designated slots and the odd digits in their designated slots. Start by calculating

\frac{4 !}{2 ! 2 !}

for the even positions.

Visualized Solution

Analyzing the Given Number

Given number: $123412341$
Total number of digits = $9$
Goal: Form $9$ -digit numbers using all these digits.

Categorizing the Digits

Odd Digits: $1, 1, 1, 3, 3$ (Total $5$ )
Even Digits: $2, 2, 4, 4$ (Total $4$ )

Identifying Even Positions

Total places = $9$
Even places: $2^{n d}, 4^{t h}, 6^{t h}, 8^{t h}$
Total even places = $4$

Understanding the Constraint

Constraint: Even digits $\to$ Even places.
Remaining odd digits $\to$ Odd places ( $1^{s t}, 3^{r d}, 5^{t h}, 7^{t h}, 9^{t h}$ ).

Arranging Even Digits

Even digits to arrange: $2, 2, 4, 4$
Formula for repeated items: $\frac{n !}{p ! q ! ...}$
Ways for even digits = $\frac{4 !}{2 ! 2 !}$

Calculating Even Arrangements

$\frac{4 !}{2 ! 2 !} = \frac{24}{2 \times 2}$
$\frac{24}{4} = 6$ ways

Arranging Odd Digits

Odd digits to arrange: $1, 1, 1, 3, 3$
Ways for odd digits = $\frac{5 !}{3 ! 2 !}$

Calculating Odd Arrangements

$\frac{5 !}{3 ! 2 !} = \frac{120}{6 \times 2}$
$\frac{120}{12} = 10$ ways

The Multiplication Principle

Total numbers = (Ways to arrange Even) $\times$ (Ways to arrange Odd)
Total = $6 \times 10 = 60$
Final Answer: $60$

00:00 / 00:00

The Sigma Insight: Linear Permutations

Solution Diagram

Analyzing the Inventory of Digits

Before we place a single digit, we must categorize our raw materials from the set

{1, 2, 3, 4, 1, 2, 3, 4, 1}

The Even Digits consist of two

2

s and two

4

s, totaling

4

digits.

The Odd Digits consist of three

1

s and two

3

s, totaling

5

digits.

The total count is

4 + 5 = 9

digits. The nine slots are divided into

4

even positions (

2^{n d}, 4^{t h}, 6^{t h}, 8^{t h}

) and

5

odd positions (

1^{s t}, 3^{r d}, 5^{t h}, 7^{t h}, 9^{t h}

The Even Constraint

The problem dictates that even digits must occupy even positions. Since we have exactly

4

even digits and

4

even slots, every even position must be filled by an even digit.

To calculate the number of arrangements, we use the permutation formula for multisets, accounting for the identical

2

s and

4

\frac{4 !}{2 ! \times 2 !} = \frac{24}{2 \times 2} = 6

There are exactly $6$ ways to arrange the even digits.

The Odd Arrangement

Next, we place the five odd digits (

1, 1, 1, 3, 3

) into the five remaining odd positions.

Again, we apply the multiset permutation formula to account for the repeated

1

s and

3

\frac{5 !}{3 ! \times 2 !} = \frac{120}{6 \times 2} = 10

There are exactly $10$ ways to arrange the odd digits in their designated slots.

The Grand Synthesis

Because the arrangement of even digits and odd digits are independent tasks, we apply the Multiplication Principle to find the total number of valid

9

-digit numbers.

Total = 6 \times 10 = 60

The total number of valid arrangements is $60$ .

Common Traps & FAQs

Wait, why are there only 4 even places? Shouldn't there be more?

In a 9-digit number, the positions are indexed 1 through 9. The even indices are 2, 4, 6, and 8. That is exactly 4 positions. Since we have exactly 4 even digits (two 2s and two 4s), they fit perfectly into these slots!

Why can't I just arrange all 9 digits and then divide by something?

That approach is risky because the constraint is positional. By separating the problem into two independent sub-problems—arranging the even digits in even slots and odd digits in odd slots—you ensure the constraint is satisfied by design rather than by chance.

Does the order of the odd digits matter if they are all 1s and 3s?

Absolutely! Even though the digits are repeated, swapping one '1' with another '1' doesn't create a new number, but swapping a '1' with a '3' does. That is why we use the formula

\frac{n !}{p ! q !}

to account for the indistinguishable nature of identical digits.

Why do we multiply the two results at the end instead of adding them?

This is the Fundamental Counting Principle. Since the arrangement of even digits and the arrangement of odd digits are independent events that must both occur to form the 9-digit number, we multiply the number of ways to perform each task.

Is it possible for an odd digit to sneak into an even place?

The problem explicitly states that even digits must occupy only even places. This implies a partition: even places are reserved for even digits, and by necessity, the remaining odd places are reserved for the remaining odd digits.

Animated Solution for Mathematics - Permutations and Combinations: The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ____.

Enter Numerical Value:

Visualized Solution

Analyzing the Given Number

Categorizing the Digits

Identifying Even Positions

Understanding the Constraint

Arranging Even Digits

Calculating Even Arrangements

Arranging Odd Digits

Calculating Odd Arrangements

The Multiplication Principle

The Sigma Insight: Linear Permutations

Analyzing the Inventory of Digits

The Even Constraint

The Odd Arrangement

The Grand Synthesis

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