Number of Ways to Arrange 1, 2, 3, 4 Such That x_i ≠ i for Every i

Understanding the Given Arrangement Problem

This question is based on a special type of permutation known as a derangement. We need to arrange the four distinct elements 1, 2, 3, and 4 in the four positions x₁, x₂, x₃, and x₄, subject to the condition:

x_i ≠ i for every i

This means that no element is allowed to occupy the position having the same number as the element itself. Therefore, element 1 cannot be placed in the first position, element 2 cannot be placed in the second position, element 3 cannot be placed in the third position, and element 4 cannot be placed in the fourth position.

In other words, if the natural arrangement is:

(1, 2, 3, 4)

then every valid arrangement must move all four elements away from their original positions. A permutation satisfying this condition is called a derangement or a permutation with no fixed points.

What Does the Condition x_i ≠ i Mean?

The sequence contains four positions:

(x₁, x₂, x₃, x₄)

The condition x_i ≠ i must hold for every position i. Therefore, the restrictions are:

x₁ ≠ 1

x₂ ≠ 2

x₃ ≠ 3

x₄ ≠ 4

For example, the sequence (2, 1, 4, 3) is valid because 2 is not in position 1, 1 is not in position 2, 4 is not in position 3, and 3 is not in position 4.

However, the sequence (2, 1, 3, 4) is not valid because x₃ = 3 and x₄ = 4. Thus, two elements remain in their original positions and the required condition fails.

Step-by-Step Solution Using the Derangement Formula

Step 1: Recognize the Problem as a Derangement

A derangement is a permutation in which no element appears in its original position. The number of derangements of n distinct elements is commonly represented by !n or D_n.

The derangement formula is:

D_n = n![1 − 1/1! + 1/2! − 1/3! + … + (−1)ⁿ/n!]

Since the present problem contains four distinct elements, we need to calculate:

D₄

Step 2: Substitute n = 4 into the Formula

Using the derangement formula:

D₄ = 4![1 − 1/1! + 1/2! − 1/3! + 1/4!]

Since:

4! = 24

we obtain:

D₄ = 24[1 − 1 + 1/2 − 1/6 + 1/24]

Step 3: Simplify the Expression

The first two terms cancel:

1 − 1 = 0

Therefore:

D₄ = 24[1/2 − 1/6 + 1/24]

Taking 24 as the common denominator:

1/2 = 12/24

1/6 = 4/24

Therefore:

D₄ = 24[(12 − 4 + 1)/24]

Thus:

D₄ = 24 × 9/24

Hence:

D₄ = 9

Solution Using the Inclusion-Exclusion Principle

The answer can also be derived from the inclusion-exclusion principle, which explains exactly why the derangement formula works.

Without any restriction, the four elements 1, 2, 3, and 4 can be arranged in:

4! = 24

different ways. From these 24 permutations, we must count only those arrangements in which none of the four elements occupies its original position.

Step 1: Count Arrangements with at Least One Fixed Element

Let A_i represent the set of permutations in which element i remains in position i. If one specified element is fixed, the remaining three elements can be arranged in:

3! = 6

ways. Since any one of the four elements may be fixed, the total initial count is:

C(4, 1) × 3! = 4 × 6 = 24

However, this subtraction counts permutations with two or more fixed elements multiple times. Therefore, the inclusion-exclusion principle requires further correction.

Step 2: Add Back Arrangements with Two Fixed Elements

If two specified elements are fixed, the remaining two elements can be arranged in:

2! = 2

ways. The two fixed elements can be chosen in:

C(4, 2) = 6

ways. Therefore, we add:

C(4, 2) × 2! = 6 × 2 = 12

Step 3: Subtract Arrangements with Three Fixed Elements

If three specified elements are fixed, the fourth element is automatically fixed as well. Algebraically, the inclusion-exclusion term is:

C(4, 3) × 1! = 4

This term must be subtracted.

Step 4: Add the Arrangement with All Four Elements Fixed

There is exactly one permutation in which all four elements remain in their original positions:

(1, 2, 3, 4)

Therefore, the final inclusion-exclusion calculation is:

D₄ = 4! − C(4, 1)3! + C(4, 2)2! − C(4, 3)1! + C(4, 4)0!

Substituting the values:

D₄ = 24 − 24 + 12 − 4 + 1

Therefore:

D₄ = 9

Direct Verification by Listing All Valid Sequences

Since there are only four elements, the result can also be verified by directly listing every valid sequence in which no element occupies its own position.

The nine valid sequences are:

(2, 1, 4, 3)
(2, 3, 4, 1)
(2, 4, 1, 3)
(3, 1, 4, 2)
(3, 4, 1, 2)
(3, 4, 2, 1)
(4, 1, 2, 3)
(4, 3, 1, 2)
(4, 3, 2, 1)

Each of these sequences satisfies all four restrictions simultaneously:

x₁ ≠ 1, x₂ ≠ 2, x₃ ≠ 3, and x₄ ≠ 4

Therefore, the direct counting method confirms that the total number of valid arrangements is exactly 9.

Alternative Solution Using the Derangement Recurrence Relation

The number of derangements can also be calculated using the recurrence relation:

D_n = (n − 1)(D_n−1 + D_n−2)

The basic derangement values are:

D₁ = 0

and:

D₂ = 1

Therefore:

D₃ = (3 − 1)(D₂ + D₁)

= 2(1 + 0)

= 2

Now:

D₄ = (4 − 1)(D₃ + D₂)

= 3(2 + 1)

= 3 × 3

Therefore:

D₄ = 9

Why This Is Not an Ordinary Permutation Problem

If there were no restriction on the positions of the elements, the total number of sequences would simply be:

4! = 24

However, the condition x_i ≠ i eliminates every arrangement containing even one fixed point. Therefore, the unrestricted value 24 cannot be used as the final answer.

The problem requires all four elements to avoid their corresponding positions simultaneously. This is precisely the defining condition of a derangement, reducing the number of valid sequences from 24 to 9.

Detailed Analysis of Each Option

Option (A): 9

This option is correct. The required arrangements are derangements of four distinct elements. Using the derangement formula:

D₄ = 4![1 − 1/1! + 1/2! − 1/3! + 1/4!]

which gives:

D₄ = 9

Direct enumeration also confirms that exactly nine sequences satisfy x_i ≠ i for every i.

Option (B): 10

This option is incorrect because the complete inclusion-exclusion calculation gives exactly nine valid permutations. A count of 10 means that at least one arrangement containing a fixed point has been incorrectly included.

Option (C): 11

This option is incorrect. The restriction applies to every index i, so any sequence in which even one element appears in its matching position must be excluded. Correct application of the derangement condition leaves only nine valid sequences.

Option (D): 12

This option is incorrect. Although 12 appears as an intermediate term in the inclusion-exclusion calculation, it is not the final number of valid arrangements. The complete calculation is:

24 − 24 + 12 − 4 + 1 = 9

Therefore, 12 cannot be the required answer.

Complete Calculation in Compact Form

The condition x_i ≠ i for every i means that the required permutations are derangements of four elements.

Therefore:

D₄ = 4![1 − 1/1! + 1/2! − 1/3! + 1/4!]

= 24[1 − 1 + 1/2 − 1/6 + 1/24]

= 24 × 9/24

= 9

Final Answer

The elements 1, 2, 3, and 4 can be arranged in 9 sequences such that x_i ≠ i for every i.

Correct Option: (A) 9