Let `S={1,2,3ddot,9}dotFork=1,2, 5,l e tN_k` be the number of subsets of S, each containing five elements out of which exactly `k` are odd. Then `N_1+N_2+N_3+N_4+N_5=?` 210 (b) 252 (c) 125 (d) 126

Options

A. 210
B. 252
C. 125
D. 126

Explanation

To solve the problem, we need to find the total number of subsets \( N_k \) of the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) that contain exactly 5 elements, out of which \( k \) are odd. We will calculate \( N_1 + N_2 + N_3 + N_4 + N_5 \).

### Step-by-Step Solution:

1. **Identify the Odd and Even Numbers:**
- The odd numbers in the set \( S \) are \( \{1, 3, 5, 7, 9\} \) (total of 5 odd numbers).
- The even numbers in the set \( S \) are \( \{2, 4, 6, 8\} \) (total of 4 even numbers).

...