There are 10 cards numbered 1 to 10 in a bag. Two cards are drawn one after other without replacement. The probability that their sum is odd, is

Options

A. `2/9`
B. `3/9`
C. `7/9`
D. `5/9`

Answer

To solve the problem, we need to find the probability that the sum of two cards drawn from a bag of 10 cards (numbered 1 to 10) is odd.

### Step-by-Step Solution:

1. **Identify the Even and Odd Numbers**:
- The cards are numbered from 1 to 10.
- The odd numbers are: 1, 3, 5, 7, 9 (total of 5 odd numbers).
- The even numbers are: 2, 4, 6, 8, 10 (total of 5 even numbers).

2. **Determine Conditions for Odd Sum**:
- The sum of two numbers is odd if one number is odd and the other is even.

3. **Calculate the Number of Favorable Outcomes**:
- To choose one odd card and one even card:
- The number of ways to choose 1 odd card from 5 odd cards is \( \binom{5}{1} = 5 \).
- The number of ways to choose 1 even card from 5 even cards is \( \binom{5}{1} = 5 \).
- Therefore, the total number of favorable outcomes (one odd and one even) is:
\[ 5 \times 5 = 25 \]

4. **Calculate the Total Number of Outcomes**:
- The total number of ways to choose any 2 cards from 10 cards is given by \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

5. **Calculate the Probability**:
- The probability \( P \) that the sum of the two drawn cards is odd is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{25}{45} = \frac{5}{9} \]

### Final Answer:
The probability that the sum of the two drawn cards is odd is \( \frac{5}{9} \).