To solve the problem of finding the probability that 6 is the third largest number when choosing 6 distinct numbers from the first 10 natural numbers, we can follow these steps:
### Step 1: Determine the Total Number of Ways to Choose 6 Numbers
The total number of ways to choose 6 distinct numbers from the first 10 natural numbers (1 to 10) is given by the combination formula:
\[
\text{Total ways} = \binom{10}{6}
\]
Calculating this:
\[
\binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]
### Step 2: Define the Event Where 6 is the Third Largest Number
For 6 to be the third largest number among the chosen 6 numbers, we need to select 2 numbers that are greater than 6 and 3 numbers that are less than 6.
### Step 3: Choose Numbers Greater than 6
The numbers greater than 6 from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} are {7, 8, 9, 10}. We need to choose 2 from these 4 numbers:
\[
\text{Ways to choose 2 from } \{7, 8, 9, 10\} = \binom{4}{2}
\]
Calculating this:
\[
\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]
### Step 4: Choose Numbers Less than 6
The numbers less than 6 from the set are {1, 2, 3, 4, 5}. We need to choose 3 from these 5 numbers:
\[
\text{Ways to choose 3 from } \{1, 2, 3, 4, 5\} = \binom{5}{3}
\]
Calculating this:
\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
\]
### Step 5: Calculate the Number of Favorable Outcomes
Now, we can find the total number of favorable outcomes where 6 is the third largest number:
\[
\text{Favorable outcomes} = \binom{4}{2} \times \binom{5}{3} = 6 \times 10 = 60
\]
### Step 6: Calculate the Probability
Finally, the probability that 6 is the third largest number is given by the ratio of the number of favorable outcomes to the total outcomes:
\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{60}{210}
\]
Simplifying this fraction:
\[
\frac{60}{210} = \frac{2}{7}
\]
### Final Answer
Thus, the probability that 6 is the third largest number among the chosen numbers is:
\[
\boxed{\frac{2}{7}}
\]
