Sara has 5 flower pots with a unique flower. The pots are arranged along a line. One day she decided to change their order the condition that no pots that were originally next to each other remain next to each other. Find out the number of possible orders satisfying the condition.
To find the number of possible orders satisfying the given condition, we can use the concept of permutations.
Let's analyze the problem step by step:
Step 1: Start with the assumption that all pots are distinct.
In this case, we have 5 distinct pots, and there are 5 possible positions for the first pot. Once the first pot is placed, there are 4 remaining pots, which can be placed in 4 possible positions. Similarly, for each subsequent pot, the number of available positions decreases by 1.
So, the total number of possible orders, assuming all pots are distinct, is:
5 × 4 × 3 × 2 × 1 = 120.
Step 2: Adjust for the condition that no pots that were originally next to each other remain next to each other.
To satisfy this condition, we need to calculate the number of cases where at least one pair of initially neighboring pots remains neighboring after rearrangement.
Let's consider the possibilities for the first two pots.
Case a: The first two pots are placed in their original order.
For this case, there's only 1 possible arrangement.
Case b: The first two pots are swapped.
For this case, there's also only 1 possible arrangement.
Now, let's calculate the total number of cases satisfying the condition:
Total cases = Total possible orders - Number of cases with at least one pair of initially neighboring pots remaining neighboring.
Total cases = 120 - (case a + case b) = 120 - (1 + 1) = 118.
Therefore, there are 118 possible orders satisfying the condition.