Geometry

Question Analysis
This question is a permutation problem with a non - adjacency constraint.
The main focus is on first finding the total number of arrangements without considering the constraint and then subtracting the number of arrangements where the restricted projects are adjacent.

Key Concept Explanation
Total permutation: Calculate the number of ways to arrange all the objects without any restrictions using the formula for distinct objects.
Adjacent - case permutation: Treat the non - allowed adjacent objects as a group to find the number of arrangements where they are together.
Then subtract this from the total permutations to get the valid arrangements.

Step - by - Step Solution
Find the total number of arrangements without any constraints:
The total number of ways to arrange 8 projects in a row is .
Find the number of arrangements where the physics and chemistry projects are adjacent:
Treat the physics and chemistry projects as one unit.
The number of ways to arrange them within this unit is (physics - chemistry or chemistry - physics).
Now, considering this unit along with the other 6 projects, we have a total of 7 units to arrange.
The number of ways to arrange these 7 units is