Question

If three cards are drawn randomly from a well-shuffled pack of \(52\) cards, then the probability that all the three bear a prime number is

• A. \(\dfrac{35}{1105}\)
• B. \(\dfrac{28}{1105}\)
• C. \(\dfrac{21}{1105}\)
• D. \(\dfrac{18}{1105}\) \bigskip

Hint:

When cards are drawn without replacement, use combinations: \[ P=\frac{\binom{\text{favourable cards}}{r}} {\binom{\text{total cards}}{r}}. \] First count the required cards carefully, then apply combinations directly.

Answer 1

The Correct Option is B

Concept: Probability = (favorable combinations) / (total combinations). Count prime cards, then use \(\binom{n}{r}\) directly and simplify the fraction.

Step 1: Identify prime-numbered cards. Primes from 2 to 10 are 2,3,5,7. With 4 suits, total = 16.

Step 2: Compute favorable outcomes. \(\binom{16}{3} = \frac{16\cdot15\cdot14}{6} = 560\).

Step 3: Compute total outcomes. \(\binom{52}{3} = \frac{52\cdot51\cdot50}{6} = 22100\).

Step 4: Form and simplify the probability. \(P = \frac{560}{22100} = \frac{56}{2210} = \frac{28}{1105}\).

Step 5: Write the final answer. \(\boxed{\frac{28}{1105}}\)