In shuffling a pack of cards three are accidentally dropped. The probability that the missing cards are of distinct suits is

Options

A. `169/425`
B. `165/429`
C. `162/459`
D. `164/529`

Answer

To solve the problem of finding the probability that the three missing cards from a shuffled pack of 52 cards are of distinct suits, we can follow these steps:

### Step 1: Determine Total Outcomes
First, we need to calculate the total number of ways to choose 3 cards from a deck of 52 cards. This can be calculated using the combination formula:

\[ \text{Total Outcomes} = \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = \frac{132600}{6} = 22100 \]

### Step 2: Determine Favorable Outcomes
Next, we need to find the number of favorable outcomes where the 3 missing cards are of distinct suits.

1. **Choosing Suits**: We have 4 suits (hearts, diamonds, clubs, spades) and we need to choose 3 out of these 4 suits. The number of ways to choose 3 suits from 4 is given by:

\[ \text{Ways to choose suits} = \binom{4}{3} = 4 \]

2. **Choosing Cards**: For each of the chosen suits, we can select 1 card. Each suit has 13 cards, so for each of the 3 chosen suits, we can choose 1 card in:

\[ \text{Ways to choose cards} = 13 \times 13 \times 13 = 13^3 = 2197 \]

3. **Total Favorable Outcomes**: Therefore, the total number of favorable outcomes where the 3 cards are of distinct suits is:

\[ \text{Favorable Outcomes} = \binom{4}{3} \times 13^3 = 4 \times 2197 = 8788 \]

### Step 3: Calculate Probability
Now we can calculate the probability that the 3 missing cards are of distinct suits. The probability is given by the ratio of favorable outcomes to total outcomes:

\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{8788}{22100} \]

### Step 4: Simplify the Probability
To simplify \(\frac{8788}{22100}\), we can find the greatest common divisor (GCD) of the numerator and the denominator. After simplification, we find:

\[ \frac{8788 \div 52}{22100 \div 52} = \frac{169}{425} \]

### Final Answer
Thus, the probability that the missing cards are of distinct suits is:

\[ \frac{169}{425} \]