the number of ways in which all the letters of the word “SWARM” can be arranged such that no letter is in its original position is

To determine the number of ways to arrange the letters of the word "SWARM" such that no letter is in its original position, we need to delve into a concept known as derangements. A derangement is a permutation of elements where none of the elements appear in their original positions. Let's break this down step by step.

Understanding Derangements

For a set of n distinct items, the number of derangements, denoted as !n, can be calculated using the formula:

• !n = n! * Σ (-1)^k / k! for k = 0 to n

In simpler terms, this formula involves calculating the factorial of n (n!) and adjusting it by summing over alternating signs of the factorial of k. This accounts for the arrangements where at least one item is in its original position, which we want to exclude.

Calculating for "SWARM"

The word "SWARM" consists of 5 distinct letters: S, W, A, R, and M. Therefore, we need to find !5.

Step-by-Step Calculation

Using the formula for derangements:

• n = 5
• 5! = 120

Now, we compute the summation:

• For k = 0: (-1)^0 / 0! = 1
• For k = 1: (-1)^1 / 1! = -1
• For k = 2: (-1)^2 / 2! = 0.5
• For k = 3: (-1)^3 / 3! = -0.1667
• For k = 4: (-1)^4 / 4! = 0.0417
• For k = 5: (-1)^5 / 5! = -0.0083

Now, summing these values:

• 1 - 1 + 0.5 - 0.1667 + 0.0417 - 0.0083 = 0.3667

Final Calculation

Now, we multiply this sum by 5!:

• !5 = 120 * 0.3667 ≈ 44

Thus, the number of ways to arrange the letters of "SWARM" such that no letter is in its original position is 44.

Conclusion

Derangements can be a fascinating topic in combinatorics, illustrating how we can count arrangements under specific constraints. In this case, we found that there are 44 unique arrangements of the letters in "SWARM" where none of the letters retains its original position.