Find the number of rearrangements of the string 12345 in which none of the sequences 12, 23, 34, 45, and 51 occur.

To find the number of rearrangements of the string "12345" such that none of the sequences "12", "23", "34", "45", and "51" appear, we can use a combinatorial approach. This problem can be tackled using the principle of inclusion-exclusion, which helps us count the valid arrangements while systematically excluding the unwanted sequences.

Understanding the Problem

The string "12345" consists of five distinct digits. We want to count the total arrangements of these digits while ensuring that certain adjacent pairs do not appear together. The pairs we want to avoid are:

• 12
• 23
• 34
• 45
• 51

Total Arrangements

First, let's calculate the total number of arrangements of the five digits without any restrictions. Since all digits are distinct, the total number of arrangements is given by:

Total arrangements = 5! = 120

Applying Inclusion-Exclusion

Next, we will apply the principle of inclusion-exclusion to subtract the arrangements that contain at least one of the unwanted pairs. We denote the sets as follows:

• A1: arrangements containing "12"
• A2: arrangements containing "23"
• A3: arrangements containing "34"
• A4: arrangements containing "45"
• A5: arrangements containing "51"

We need to find the size of the union of these sets, |A1 ∪ A2 ∪ A3 ∪ A4 ∪ A5|. Using inclusion-exclusion, we have:

|A1 ∪ A2 ∪ A3 ∪ A4 ∪ A5| = Σ|Ai| - Σ|Ai ∩ Aj| + Σ|Ai ∩ Aj ∩ Ak| - ...

Calculating |Ai| for Each Pair

For any single pair, say "12", we can treat it as a single unit or block. Thus, the arrangements of the blocks "12", "3", "4", "5" can be calculated as follows:

|A1| = 4! = 24

This same logic applies to each of the pairs, so:

|A1| = |A2| = |A3| = |A4| = |A5| = 24

Calculating |Ai ∩ Aj| for Pairs

Next, we consider intersections of two pairs, such as "12" and "23". If both are present, we can treat "12" and "23" as blocks, leading to the arrangement of "12", "3", "4", "5" as a single unit. The arrangements can be calculated as:

|A1 ∩ A2| = 3! = 6

By symmetry, this holds for all pairs of adjacent pairs, leading to:

|Ai ∩ Aj| = 6 for each pair of adjacent pairs.

Counting All Intersections

For three pairs, such as "12", "23", and "34", we treat them as blocks, leading to arrangements of "12", "3", "4", "5" as:

|A1 ∩ A2 ∩ A3| = 2! = 2

Continuing this logic, we can find that:

• |A1 ∩ A2 ∩ A3| = |A2 ∩ A3 ∩ A4| = |A3 ∩ A4 ∩ A5| = 2
• |A1 ∩ A2 ∩ A3 ∩ A4| = |A2 ∩ A3 ∩ A4 ∩ A5| = 1

Final Calculation

Now, we can plug these values into our inclusion-exclusion formula:

|A1 ∪ A2 ∪ A3 ∪ A4 ∪ A5| = 5 * 24 - 10 * 6 + 10 * 2 - 5 * 1 = 120 - 60 + 20 - 5 = 75

Valid Arrangements

Finally, to find the number of valid arrangements that do not contain any of the unwanted pairs, we subtract the count of invalid arrangements from the total arrangements:

Valid arrangements = Total arrangements - |A1 ∪ A2 ∪ A3 ∪ A4 ∪ A5| = 120 - 75 = 45

Thus, the number of rearrangements of the string "12345" in which none of the sequences "12", "23", "34", "45", and "51" occur is 45.