Question

An integer is chosen at random from the integers 1,2, 3, ..., 50. The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is

• A. \( \frac{8}{25} \)
• B. \( \frac{21}{50} \)
• C. \( \frac{9}{50} \)
• D. \( \frac{14}{25} \)

Answer 1

The Correct Option is B

To determine the probability that a randomly selected integer from 1 to 50 is a multiple of 4, 6, or 7, the Principle of Inclusion-Exclusion will be applied. The process is as follows:

1. Determine the count of multiples for each individual number within the range 1 to 50:
   • Multiples of 4: These are 4, 8, ..., 48. The count is \( 48 \div 4 = 12 \).
   • Multiples of 6: These are 6, 12, ..., 48. The count is \( 48 \div 6 = 8 \).
   • Multiples of 7: These are 7, 14, ..., 49. The count is \( 49 \div 7 = 7 \).
2. Determine the count of multiples for the least common multiples (LCM) of pairs of numbers:
   • LCM(4, 6) = 12: Multiples are 12, 24, 36, 48. The count is \( 48 \div 12 = 4 \).
   • LCM(6, 7) = 42: The only multiple within the range is 42. The count is \( 50 \div 42 = 1 \).
   • LCM(4, 7) = 28: The only multiple within the range is 28. The count is \( 50 \div 28 = 1 \).
3. Determine the count of multiples for the LCM of all three numbers:
   • LCM(4, 6, 7) = 84: There are no multiples of 84 within the range 1 to 50. The count is \( 50 \div 84 = 0 \).
4. Apply the Inclusion-Exclusion principle: The number of integers that are multiples of at least one of 4, 6, or 7 is calculated as (sum of individual counts) - (sum of pairwise LCM counts) + (count of LCM of all three). 
   Number of multiples = \( 12 + 8 + 7 - 4 - 1 - 1 + 0 = 21 \).
5. Calculate the probability: The total number of integers is 50. 
   The probability \( P \) is the number of favorable outcomes divided by the total number of outcomes. 
   Probability \( P \) = \(\frac{21}{50}\).

The calculated probability is \(\frac{21}{50}\).