Question

If three cards are drawn randomly from a pack of 52 playing cards then the probability of getting exactly one spade card, exactly one king and exactly one card having a prime number is

• A. \( \frac{72}{221} \)
• B. \( \frac{72}{5525} \)
• C. \( \frac{16}{425} \)
• D. \( \frac{144}{5525} \)

Hint:

When requirements like "exactly one" are combined with overlapping sets (like King and Spade), defining disjoint subsets for selection is crucial to avoid double-counting or ambiguity.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

We need to select three cards that satisfy specific, potentially overlapping properties. However, based on the options, the problem implies selecting distinct cards for each role (a Spade, a King, a Prime) without overlap in their roles for this specific counts calculation.
Step 2: Key Formula or Approach:

\[ P(E) = \frac{\text{Ways to choose favorable cards}}{\text{Total ways to choose 3 cards}} \] We define disjoint sets for "Spade only", "King only", and "Prime only" to match the criteria.
Step 3: Detailed Explanation:

Let's define the sets of cards based on the requirements: 1. One Spade:
Must be a Spade, but not a King, and not a Prime (to ensure "exactly one" of each type in the selection set). - Spades: 13 cards. - King of Spades: 1 card. - Primes in Spades (2, 3, 5, 7): 4 cards. - Available Spades = \( 13 - 1 - 4 = 8 \). 2. One King:
Must be a King, but not a Spade. - Kings: 4 cards. - King of Spades: 1 (already excluded). - Available Kings = \( 4 - 1 = 3 \). 3. One Prime:
Must be a Prime, but not a Spade. - Primes: \{2, 3, 5, 7\} in each suit. - Total Primes = \( 4 \times 4 = 16 \). - Primes in Spades: 4 (already excluded). - Available Primes = \( 16 - 4 = 12 \). Number of ways to choose one of each: \[ n(E) = \binom{8}{1} \times \binom{3}{1} \times \binom{12}{1} = 8 \times 3 \times 12 = 288 \] Total ways to choose 3 cards: \[ n(S) = \binom{52}{3} = \frac{52 \times 51 \times 50}{6} = 22100 \] Probability: \[ P(E) = \frac{288}{22100} = \frac{72}{5525} \]
Step 4: Final Answer:

The probability is \( \frac{72}{5525} \).