Question

While shuffling a pack of cards, 3 cards were accidentally dropped, then find the probability that the missing cards belong to different suits?

• A. \( \frac{104}{425} \)
• B. \( \frac{169}{425} \)
• C. \( \frac{261}{425} \)
• D. \( \frac{169}{261} \)

Hint:

To solve probability problems with multiple outcomes, first calculate the total number of possible outcomes and then determine the number of favorable outcomes by applying combinations.

Answer 1

The Correct Option is B

A standard pack of cards contains 52 cards, divided into 4 suits of 13 cards each. We are interested in finding the probability that the 3 cards that are dropped belong to different suits. Total number of ways to choose 3 cards from 52 cards: \[ \text{Total ways} = \binom{52}{3} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22100 \] Now, we need to find the number of favorable outcomes where the 3 cards belong to different suits. To do this: 1. Choose 3 different suits from 4 suits: \( \binom{4}{3} = 4 \) 2. For each chosen suit, select 1 card (13 options for each suit): \[ \text{Ways to select cards from different suits} = 13 \times 13 \times 13 = 13^3 = 2197 \] Thus, the number of favorable outcomes is: \[ \text{Favorable outcomes} = 4 \times 13^3 = 4 \times 2197 = 8788 \] Now, the probability that the 3 cards belong to different suits is the ratio of favorable outcomes to total outcomes: \[ P = \frac{8788}{22100} \] \[ P = \frac{169}{425} \] Thus, the correct answer is (B).