If 3 distinct numbers are chosen randomly from {1,2, . .. 100}, then probability that all are divisible by both 2 and 3 is

Options

A. `4//25`
B. `4//35`
C. `4//33`
D. `4//1155`

Answer

To find the probability that all three distinct numbers chosen randomly from the set {1, 2, ..., 100} are divisible by both 2 and 3, we can follow these steps:

### Step 1: Determine the total number of outcomes
The total number of ways to choose 3 distinct numbers from 100 numbers is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.

\[ \text{Total outcomes} = \binom{100}{3} \]

### Step 2: Find the favorable outcomes
Next, we need to determine how many numbers between 1 and 100 are divisible by both 2 and 3. The least common multiple (LCM) of 2 and 3 is 6. Thus, we need to count how many multiples of 6 are there in the range from 1 to 100.

The multiples of 6 up to 100 can be found by calculating:

\[ \text{Largest multiple of 6} \leq 100 = 6 \times 16 = 96 \]

This means the multiples of 6 from 1 to 100 are: 6, 12, 18, ..., 96. The number of multiples of 6 is given by:

\[ \text{Number of multiples of 6} = \frac{96}{6} = 16 \]

### Step 3: Calculate the number of favorable outcomes
Now, we need to choose 3 distinct numbers from these 16 multiples of 6:

\[ \text{Favorable outcomes} = \binom{16}{3} \]

### Step 4: Calculate the probability
The probability \( P \) that all three chosen numbers are divisible by both 2 and 3 is given by the ratio of favorable outcomes to total outcomes:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\binom{16}{3}}{\binom{100}{3}} \]

### Step 5: Compute the combinations
Now we compute \( \binom{16}{3} \) and \( \binom{100}{3} \):

\[ \binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = \frac{3360}{6} = 560 \]

\[ \binom{100}{3} = \frac{100 \times 99 \times 98}{3 \times 2 \times 1} = \frac{970200}{6} = 161700 \]

### Step 6: Final probability calculation
Now substituting back into the probability formula:

\[ P = \frac{560}{161700} \]

### Step 7: Simplify the fraction
To simplify \( \frac{560}{161700} \):

1. Find the GCD of 560 and 161700, which is 20.
2. Divide both the numerator and the denominator by 20:

\[ P = \frac{28}{8085} \]

Thus, the final probability that all three chosen numbers are divisible by both 2 and 3 is:

\[ \frac{28}{8085} \]