Question

Two numbers P and Q are selected from Set {1,2,3,.....,100\,then find the probability of P.Q is divisible by 5.}

• A. \(\frac{189}{495}\)
• B. \(\frac{79}{495}\)
• C. \(\frac{179}{495}\)
• D. \(\frac{9}{495}\)

Hint:

Whenever a question asks for "at least one" condition to be met (like product divisible by a prime meaning at least one factor is divisible), calculating the probability of the opposite event (none) is usually much simpler.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are selecting two distinct numbers (since it's a typical selection from a set without replacement unless stated otherwise) from 1 to 100. We need the probability that their product is divisible by 5. A product is divisible by 5 if and only if at least one of the numbers is a multiple of 5.


Step 2: Key Formula or Approach:
Probability = \(\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}\)
P(at least one multiple of 5) = 1 - P(no multiple of 5)
Or, calculate favorable outcomes by subtracting non-favorable outcomes from the total.


Step 3: Detailed Explanation:
Total numbers in the set = 100.
Total ways to select 2 distinct numbers = \(\binom{100}{2} = \frac{100 \times 99}{2} = 4950\).
Let's find the number of non-favorable outcomes, i.e., when NEITHER P nor Q is divisible by 5.
Number of multiples of 5 in the set \(\{1, 2, \dots, 100\}\) = \(100 / 5 = 20\).
Number of non-multiples of 5 = \(100 - 20 = 80\).
Ways to select 2 numbers such that neither is a multiple of 5 = \(\binom{80}{2}\)
\[ \binom{80}{2} = \frac{80 \times 79}{2} = 40 \times 79 = 3160 \] Now, calculate the number of favorable outcomes (at least one multiple of 5): \[ \text{Favorable ways} = \text{Total ways} - \text{Non-favorable ways} \] \[ \text{Favorable ways} = 4950 - 3160 = 1790 \] The required probability is: \[ P = \frac{1790}{4950} = \frac{179}{495} \]

Step 4: Final Answer:
The probability is \(\frac{179}{495}\).