Permutations with Fixed Points

To solve this problem, we need to find the number of permutations of the set {1, 2, 3, 4, 5} that keep at least one integer fixed.

Let's consider the different possibilities for the number of fixed points in a permutation:

1. One fixed point: In this case, we can choose any of the 5 integers to be the fixed point. Once we fix one integer, we have 4 remaining integers that can be arranged in any order. Therefore, there are 5 * 4! = 120 permutations with one fixed point.

2. Two fixed points: To calculate the number of permutations with two fixed points, we need to consider all possible pairs of fixed points. There are 5C2 = 10 ways to choose two integers from the set {1, 2, 3, 4, 5}. Once we fix two integers, we have 3 remaining integers that can be arranged in any order. Therefore, there are 10 * 3! = 60 permutations with two fixed points.

3. Three fixed points: To calculate the number of permutations with three fixed points, we need to consider all possible triples of fixed points. There are 5C3 = 10 ways to choose three integers from the set {1, 2, 3, 4, 5}. Once we fix three integers, we have 2 remaining integers that can be arranged in any order. Therefore, there are 10 * 2! = 20 permutations with three fixed points.

4. Four fixed points: To calculate the number of permutations with four fixed points, we need to consider all possible quadruples of fixed points. There are 5C4 = 5 ways to choose four integers from the set {1, 2, 3, 4, 5}. Once we fix four integers, we have 1 remaining integer that can be arranged in any order. Therefore, there are 5 * 1! = 5 permutations with four fixed points.

5. Five fixed points: In this case, all integers are fixed and there is only one possible permutation.

Total number of permutations with at least one fixed point:
To get the total number of permutations with at least one fixed point, we sum up the number of permutations for each case:

Total = 120 + 60 + 20 + 5 + 1 = 206

Therefore, the correct answer is option 'B' (76 permutations).