Three distinct number are selected from first 100 natural numbers. The probability that all the numbers are divisible by 2 and 3 is _________

Answer

To solve the problem of finding the probability that three distinct numbers selected from the first 100 natural numbers are all divisible by both 2 and 3, we can follow these steps:

### Step 1: Understand the Problem
We need to find the probability that all selected numbers are divisible by both 2 and 3. Since a number divisible by both 2 and 3 is also divisible by 6, we will focus on numbers divisible by 6.

### Step 2: Identify the Total Number of Natural Numbers
The first 100 natural numbers are from 1 to 100. The total number of ways to select 3 distinct numbers from these 100 numbers is given by the combination formula:
\[ \text{Total ways} = \binom{100}{3} = \frac{100 \times 99 \times 98}{3 \times 2 \times 1} \]

### Step 3: Count the Numbers Divisible by 6
Next, we find how many numbers between 1 and 100 are divisible by 6. The sequence of numbers divisible by 6 is:
6, 12, 18, ..., 96.

To find the largest number less than or equal to 100 that is divisible by 6, we can use:
\[ \text{Largest number} = 6 \times n \quad \text{where } n \text{ is the largest integer such that } 6n \leq 100. \]
Calculating \(n\):
\[ n = \left\lfloor \frac{100}{6} \right\rfloor = 16. \]
Thus, there are 16 numbers (6, 12, 18, ..., 96) that are divisible by 6.

### Step 4: Calculate the Ways to Select 3 Distinct Numbers from Those Divisible by 6
Now we need to find the number of ways to select 3 distinct numbers from these 16 numbers:
\[ \text{Ways to select 3 numbers} = \binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1}. \]

### Step 5: Calculate the Probability
The probability that all three selected numbers are divisible by 6 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Ways to select 3 numbers divisible by 6}}{\text{Total ways to select 3 numbers}} = \frac{\binom{16}{3}}{\binom{100}{3}}. \]

### Step 6: Substitute and Simplify
Substituting the values we calculated:
\[ \text{Total ways} = \binom{100}{3} = \frac{100 \times 99 \times 98}{6}, \]
\[ \text{Ways to select 3 numbers divisible by 6} = \binom{16}{3} = \frac{16 \times 15 \times 14}{6}. \]
Thus, the probability becomes:
\[ \text{Probability} = \frac{\frac{16 \times 15 \times 14}{6}}{\frac{100 \times 99 \times 98}{6}} = \frac{16 \times 15 \times 14}{100 \times 99 \times 98}. \]

### Step 7: Final Calculation
Now we can calculate the final probability:
\[ \text{Probability} = \frac{16 \times 15 \times 14}{100 \times 99 \times 98}. \]
Calculating this gives:
\[ \text{Probability} = \frac{3360}{970200} = \frac{4}{1155}. \]

### Final Answer
The probability that all three selected numbers are divisible by 2 and 3 is:
\[ \frac{4}{1155}. \]