A person draws 2 cards from a well shuffled pack of cards, the cards are replaced after noting their colour. Then another person draws 2 cards after shuffling the pack. The probability that there will be exactly 1 common card is

Options

A. `(25)/(546)`
B. `(50)/(663)`
C. `(25)/(663)`
D. none of these

Explanation

To solve the problem of finding the probability that there will be exactly 1 common card when two people draw cards from a well-shuffled deck, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- The first person draws 2 cards from a deck of 52 cards, notes their colors, and replaces them.
- The second person then draws 2 cards from the same deck after shuffling it again.
- We need to find the probability that exactly 1 card is common between the two draws.

2. **Total Number of Ways to Draw Cards**:
- The total number of ways to choose 2 cards from 52 is given by the combination formula \( C(n, k) \), which is \( C(52, 2) = \frac{52 \times 51}{2} = 1326 \).

3. **Choosing the Common Card**:
- We can choose 1 common card from the 2 cards drawn by the first person. There are 2 ways to choose which of the 2 cards will be the common card.

4. **Choosing the Second Card for Each Person**:
- After choosing 1 common card, the first person has already drawn 2 cards, so they have 1 remaining card that is not common.
- The second person must draw a card that is not the common card and not the other card drawn by the first person. There are \( 52 - 2 = 50 \) cards left for the second person to choose from.

5. **Calculating the Probability**:
- The probability that the second person draws 1 common card and 1 different card can be calculated as follows:
- The number of favorable outcomes for the second person is \( C(1, 1) \times C(50, 1) = 1 \times 50 = 50 \).
- The total outcomes for the second person drawing 2 cards is \( C(52, 2) = 1326 \).
- Therefore, the probability \( P \) that there is exactly 1 common card is:
\[ P = \frac{2 \times 50}{1326} = \frac{100}{1326} = \frac{50}{663} \]

6. **Final Answer**:
- The probability that there will be exactly 1 common card is \( \frac{50}{663} \).