From numbers 1 to 15 two no are selected what is the probability that both the selected no . Are even numbers.

Options

A. `0.20`
B. `0.30`
C. `0.25`
D. `0.35`

Answer

To solve the problem of finding the probability that both selected numbers from 1 to 15 are even numbers, we can follow these steps:

### Step 1: Identify the Total Numbers
The total numbers from 1 to 15 are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \]
Thus, the total count of numbers is:
\[ \text{Total Numbers} = 15 \]

**Hint:** Count all the integers from 1 to 15 to find the total.

### Step 2: Identify the Even Numbers
The even numbers between 1 and 15 are:
\[ 2, 4, 6, 8, 10, 12, 14 \]
Thus, the total count of even numbers is:
\[ \text{Total Even Numbers} = 7 \]

**Hint:** List out the even numbers in the given range to find how many there are.

### Step 3: Calculate the Total Ways to Select 2 Numbers
The total ways to select 2 numbers from 15 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

For our case:
\[ n = 15, \quad r = 2 \]
So,
\[ \text{Total Ways to Select 2 Numbers} = \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 \]

**Hint:** Use the combination formula to find the number of ways to choose 2 items from a larger set.

### Step 4: Calculate the Ways to Select 2 Even Numbers
Now, we calculate the ways to select 2 even numbers from the 7 even numbers:
\[ \text{Ways to Select 2 Even Numbers} = \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

**Hint:** Again, use the combination formula to find the number of ways to choose 2 even numbers.

### Step 5: Calculate the Probability
The probability \( P \) that both selected numbers are even is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{both even}) = \frac{\text{Ways to Select 2 Even Numbers}}{\text{Total Ways to Select 2 Numbers}} = \frac{21}{105} \]
Simplifying this fraction:
\[ P(\text{both even}) = \frac{1}{5} \]

**Hint:** The probability is found by dividing the number of successful outcomes by the total outcomes.

### Final Answer
Thus, the probability that both selected numbers are even is:
\[ \boxed{\frac{1}{5}} \]