Five persons A, B, C, D & E are seated in a circular arrangement. If each of the is given a hat of one of the three colours red, blue & green, then the numbers of ways of distributing the hats such that the person seated in adjacent seat gets different coloured hats is

Explanation

To solve the problem of distributing hats of three colors (red, blue, and green) to five persons seated in a circular arrangement such that no two adjacent persons wear the same color hat, we can follow these steps:

### Step 1: Understand the Problem
We have five persons (A, B, C, D, and E) seated in a circle. Each person can wear a hat of one of three colors: red, blue, or green. The condition is that adjacent persons cannot wear hats of the same color.

### Step 2: Use the Chromatic Polynomial
This problem can be approached using the concept of the chromatic polynomial for a cycle graph. The number of ways to color a cycle graph \( C_n \) with \( k \) colors such that no two adjacent vertices (persons) share the same color is given by the formula:
\[ P(C_n, k) = (k - 1)^n + (-1)^n (k - 1) \]
where \( n \) is the number of vertices (persons) and \( k \) is the number of colors.

### Step 3: Apply the Formula
In our case, we have:
- \( n = 5 \) (the number of persons)
- \( k = 3 \) (the number of colors)

Substituting these values into the formula:
\[ P(C_5, 3) = (3 - 1)^5 + (-1)^5 (3 - 1) \]
Calculating this:
\[ P(C_5, 3) = 2^5 - 2 = 32 - 2 = 30 \]

### Step 4: Conclusion
Thus, the total number of ways to distribute the hats such that no two adjacent persons have the same color hat is **30**.