To find the number of ways in which three numbers in an arithmetic progression (A.P.) can be selected from the set {1, 2, 3, ..., n}, where n is odd, we can proceed as follows:

1. Determining the common difference (d):
Since the numbers are in an arithmetic progression, there is a common difference between consecutive terms. We can find this common difference by subtracting the first term (1) from the second term (2), which gives us d = 2 - 1 = 1.

2. Identifying the maximum possible value for the middle term (m):
Since the set contains n elements and n is odd, the middle term can be found by taking the average of the first and last terms. So, m = (1 + n) / 2.

3. Counting the number of possible values for the middle term:
Since the middle term must be an integer, we need to consider whether m is an integer or a fraction. If m is an integer, then there is only one possible value for the middle term. However, if m is a fraction, then we have two possible values for the middle term.

4. Calculating the number of ways to select three numbers in A.P.:
If m is an integer, then we have one possible value for the middle term. In this case, the number of ways to select three numbers in A.P. is equal to the number of ways to select two numbers from the set {1, 2, ..., m-1} multiplied by the number of ways to select one number from the set {m+1, m+2, ..., n}.

If m is a fraction, then we have two possible values for the middle term. In this case, the number of ways to select three numbers in A.P. is equal to the number of ways to select one number from the set {1, 2, ..., m-1} multiplied by the number of ways to select one number from the set {m+1, m+2, ..., n} multiplied by 2, since we have two possible middle terms.

5. Simplifying the expression:
Using the formula for the sum of an arithmetic series, we can simplify the expression for the number of ways to select three numbers in A.P. as follows:

Number of ways = (m-1) * (n-m) + 2 * (m-1) * (n-m)
= (m-1) * [ (n-m) + 2 * (n-m) ]
= (m-1) * (n-m+2n-2m)
= (m-1) * (3n-3m)

6. Substituting the value of m:
Since m = (1 + n) / 2, we can substitute this value into the expression derived in step 5:

Number of ways = [(1 + n) / 2 - 1] * [3n - 3((1 + n) / 2)]
= [(1 + n - 2) / 2] * [3n - 3 - 3(n + 1) / 2]
= [n - 1] / 2 * [3n - 3 - (3n + 3) / 2]
= [n - 1] /