Two numbers are selected from first 40 natural numbers. The probability that the sum of two number is odd is:

Options

A. `16/39`
B. `19/39`
C. `20/39`
D. None of these

Explanation

To solve the problem of finding the probability that the sum of two numbers selected from the first 40 natural numbers is odd, we can follow these steps:

### Step 1: Understand the conditions for the sum to be odd
The sum of two numbers is odd if one number is odd and the other is even.

### Step 2: Identify the odd and even numbers in the first 40 natural numbers
In the first 40 natural numbers:
- Odd numbers: 1, 3, 5, ..., 39 (total of 20 odd numbers)
- Even numbers: 2, 4, 6, ..., 40 (total of 20 even numbers)

### Step 3: Calculate the total number of ways to choose 2 numbers from 40
The total number of ways to choose 2 numbers from 40 is given by the combination formula:
\[ \text{Total ways} = \binom{40}{2} = \frac{40 \times 39}{2} = 780 \]

### Step 4: Calculate the number of favorable outcomes
To have an odd sum, we need to choose one odd number and one even number. The number of ways to choose one odd number from 20 odd numbers and one even number from 20 even numbers is:
\[ \text{Favorable outcomes} = \binom{20}{1} \times \binom{20}{1} = 20 \times 20 = 400 \]

### Step 5: Calculate the probability
The probability \( P \) that the sum of the two selected numbers is odd can be calculated as:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{400}{780} \]

### Step 6: Simplify the fraction
To simplify \( \frac{400}{780} \):
\[ P = \frac{400 \div 20}{780 \div 20} = \frac{20}{39} \]

### Final Answer
Thus, the probability that the sum of the two selected numbers is odd is:
\[ \boxed{\frac{20}{39}} \]

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