Problem:
Let S = {1, 2, 3,.....,9}. For k = 1, 2, ....., 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Find N1, N2, N3, N5.

Solution:
To solve this problem, we can divide it into smaller sub-problems and count the number of subsets for each case.

Case 1: Exactly 1 odd number in the subset (k = 1)
To have exactly 1 odd number in the subset, we have to choose 1 odd number from the set of odd numbers (1, 3, 5, 7, 9) and 4 even numbers from the set of even numbers (2, 4, 6, 8).
Number of ways to choose 1 odd number from a set of 5 odd numbers = 5C1 = 5
Number of ways to choose 4 even numbers from a set of 4 even numbers = 4C4 = 1
Total number of subsets = 5 x 1 = 5
So, N1 = 5.

Case 2: Exactly 2 odd numbers in the subset (k = 2)
To have exactly 2 odd numbers in the subset, we have to choose 2 odd numbers from the set of odd numbers and 3 even numbers from the set of even numbers.
Number of ways to choose 2 odd numbers from a set of 5 odd numbers = 5C2 = 10
Number of ways to choose 3 even numbers from a set of 4 even numbers = 4C3 = 4
Total number of subsets = 10 x 4 = 40
So, N2 = 40.

Case 3: Exactly 3 odd numbers in the subset (k = 3)
To have exactly 3 odd numbers in the subset, we have to choose 3 odd numbers from the set of odd numbers and 2 even numbers from the set of even numbers.
Number of ways to choose 3 odd numbers from a set of 5 odd numbers = 5C3 = 10
Number of ways to choose 2 even numbers from a set of 4 even numbers = 4C2 = 6
Total number of subsets = 10 x 6 = 60
So, N3 = 60.

Case 4: Exactly 5 odd numbers in the subset (k = 5)
To have exactly 5 odd numbers in the subset, we have to choose 5 odd numbers from the set of odd numbers and no even numbers from the set of even numbers.
Number of ways to choose 5 odd numbers from a set of 5 odd numbers = 5C5 = 1
Number of ways to choose 0 even numbers from a set of 4 even numbers = 4C0 = 1
Total number of subsets = 1 x 1 = 1
So, N5 = 1.

Summary:
N1 = 5
N2 = 40
N3 = 60
N5 = 1

Therefore, the correct answer is option D) 126