Question

If three distinct numbers are chosen randomly from the first 50 natural numbers, then the probability that all of them are divisible by 2 and 3 is:

• A. \( \frac{3}{350} \)
• B. 3
• C. \( \frac{2}{175} \)
• D. \( \frac{1}{175} \)
• E. \( \frac{1}{350} \)

Hint:

To find the probability, calculate the favorable outcomes and divide by the total outcomes. Use combinations for selecting distinct numbers.

Answer 1

Answer: (E)

The numbers divisible by both 2 and 3 are divisible by 6. So, we need to find how many numbers from 1 to 50 are divisible by 6. 
These numbers are: \[ 6, 12, 18, 24, 30, 36, 42, 48 \] Thus, there are 8 numbers divisible by 6. To choose 3 distinct numbers from these 8, the number of ways is: \[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \] The total number of ways to choose 3 distinct numbers from 50 is: \[ \binom{50}{3} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 19600 \] Therefore, the probability is: \[ \frac{56}{19600} = \frac{1}{350} \]