Problem Analysis:
We are given that x denotes the number of times heads occur in n tosses of a fair coin. We need to find the value of n when the probabilities P(x=4), P(x=5), and P(x=6) are in arithmetic progression (AP).

Solution:

Step 1: Understanding the Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, the terms of an arithmetic progression form a sequence in which the difference between consecutive terms is always the same.

Step 2: Understanding the Probability Distribution
In this problem, x represents the number of times heads occur in n tosses of a fair coin. Since it is a fair coin, the probability of getting a head on each toss is 0.5, and the probability of getting a tail is also 0.5. Therefore, the probability distribution of x follows a binomial distribution.

Step 3: Finding the Probabilities
To find the probabilities P(x=4), P(x=5), and P(x=6), we need to use the binomial probability formula:

P(x) = (nC x) * (p^x) * (q^(n-x))

Where nC x represents n choose x (the number of ways to choose x items from a set of n items), p represents the probability of getting a head on a single toss, q represents the probability of getting a tail on a single toss, and x represents the number of times heads occur.

Since it is a fair coin, p = q = 0.5.

Using the formula, we can calculate the probabilities as follows:

P(x=4) = (nC4) * (0.5^4) * (0.5^(n-4))
P(x=5) = (nC5) * (0.5^5) * (0.5^(n-5))
P(x=6) = (nC6) * (0.5^6) * (0.5^(n-6))

Step 4: Forming an Arithmetic Progression
We are given that P(x=4), P(x=5), and P(x=6) are in arithmetic progression.

Let the common difference be d.

P(x=5) - P(x=4) = d
P(x=6) - P(x=5) = d

Using the formulas derived earlier, we can substitute the values and solve for n:

[(nC5) * (0.5^5) * (0.5^(n-5))] - [(nC4) * (0.5^4) * (0.5^(n-4))] = d
[(nC6) * (0.5^6) * (0.5^(n-6))] - [(nC5) * (0.5^5) * (0.5^(n-5))] = d

Simplifying these equations will give us a relationship between n and d.

Step 5: Solving for n
To solve for n, we can substitute d = P(x=5) - P(x=4) into the equation [(nC5)