While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of different colours.

Options

A. `(29)/(52)`
B. `(1)/(2)`
C. `(26)/(51)`
D. `(27)/(51)`

Explanation

To solve the problem of finding the probability that the two missing cards from a pack of 52 playing cards are of different colors, we can follow these steps:

### Step 1: Understand the Composition of the Deck
A standard deck of 52 playing cards consists of:
- 26 red cards (hearts and diamonds)
- 26 black cards (clubs and spades)

### Step 2: Total Ways to Choose 2 Cards
The total number of ways to choose 2 cards from the 52 cards is given by the combination formula:
\[ \text{Total ways} = \binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 1326 \]

### Step 3: Calculate Favorable Outcomes
To find the number of ways to choose 2 cards of different colors, we can choose 1 red card and 1 black card. The number of ways to choose 1 red card from 26 red cards and 1 black card from 26 black cards is:
\[ \text{Ways to choose 1 red and 1 black} = \binom{26}{1} \times \binom{26}{1} = 26 \times 26 = 676 \]

### Step 4: Calculate the Probability
The probability that the two missing cards are of different colors is the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{676}{1326} \]

### Step 5: Simplify the Probability
We can simplify the fraction:
\[ \frac{676}{1326} = \frac{338}{663} = \frac{169}{331} \]

Thus, the probability that the two missing cards are of different colors is:
\[ \text{Probability} = \frac{169}{331} \]

### Final Answer
The probability that the missing cards are of different colors is \(\frac{169}{331}\).

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