If 2 distinct numbers are between 0 to 180 (both inclusive) and the probability that their average is 60 is k, then 1086k is equal to

Answer

To solve the problem, we need to find the probability \( k \) that the average of two distinct numbers chosen from the range [0, 180] is 60, and then calculate \( 1086k \).

### Step-by-Step Solution:

1. **Determine the total number of distinct numbers:**
The numbers range from 0 to 180, inclusive. Therefore, the total count of numbers is:
\[ 180 - 0 + 1 = 181 \]

2. **Calculate the total ways to choose 2 distinct numbers:**
The number of ways to choose 2 distinct numbers from 181 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total ways} = \binom{181}{2} = \frac{181 \times 180}{2} = 16290 \]

3. **Set up the condition for the average:**
If we denote the two distinct numbers as \( x_1 \) and \( x_2 \), their average being 60 implies:
\[ \frac{x_1 + x_2}{2} = 60 \implies x_1 + x_2 = 120 \]

4. **Identify valid pairs \( (x_1, x_2) \):**
We can express \( x_2 \) in terms of \( x_1 \):
\[ x_2 = 120 - x_1 \]
To ensure both \( x_1 \) and \( x_2 \) are within the range [0, 180], we need:
- \( 0 \leq x_1 < 120 \) (since \( x_1 \) must be distinct from \( x_2 \))
- \( x_2 \) must also be distinct from \( x_1 \), which means \( 120 - x_1 \neq x_1 \) or \( x_1 \neq 60 \).

5. **Count the valid pairs:**
The valid values for \( x_1 \) range from 0 to 119 (i.e., 120 possible values). However, since \( x_1 \) cannot be 60 (to ensure distinctness), we have:
\[ 120 - 1 = 119 \text{ valid choices for } x_1. \]

6. **Calculate the number of valid pairs:**
Each valid \( x_1 \) corresponds to a unique \( x_2 \) (since \( x_2 = 120 - x_1 \)), hence the number of valid pairs is:
\[ 119 \]

7. **Calculate the probability \( k \):**
The probability \( k \) that the average of the two distinct numbers is 60 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ k = \frac{119}{16290} \]

8. **Calculate \( 1086k \):**
Now we compute \( 1086k \):
\[ 1086k = 1086 \times \frac{119}{16290} \]
Simplifying this:
\[ 1086k = \frac{1086 \times 119}{16290} \]
We can simplify \( \frac{1086}{16290} \):
\[ 1086 = 2 \times 543, \quad 16290 = 90 \times 181 \]
Thus:
\[ 1086k = \frac{119 \times 2}{90} = \frac{238}{90} = \frac{119}{45} \]

9. **Final Calculation:**
The final answer is:
\[ 1086k = 4 \]