Two cards are drawn at random from a well shuffled pack of 52 cards. The probability that cards are of different suits is a) `1/221` b) `2/221` c) `39/51` d) `25/52`

Options

A. a) `1/221`
B. b) `2/221`
C. c) `39/51`
D. d) `25/52`

Answer

To find the probability that two cards drawn from a well-shuffled pack of 52 cards are of different suits, we can follow these steps:

### Step 1: Determine the total number of cards and suits
- A standard pack of cards has 52 cards divided into 4 suits: Spades, Clubs, Diamonds, and Hearts. Each suit has 13 cards.

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

### Step 3: Calculate the number of ways to choose 2 cards of different suits
- To find the number of ways to choose 2 cards of different suits, we can first choose 2 different suits from the 4 available suits. The number of ways to choose 2 suits from 4 is:
\[ \text{Ways to choose suits} = \binom{4}{2} = 6 \]
- For each pair of suits chosen, we can choose 1 card from each suit. The number of ways to choose 1 card from each of the 2 suits (each having 13 cards) is:
\[ \text{Ways to choose cards} = 13 \times 13 = 169 \]
- Therefore, the total number of ways to choose 2 cards of different suits is:
\[ \text{Total ways for different suits} = 6 \times 169 = 1014 \]

### Step 4: Calculate the probability
- The probability that the two cards drawn are of different suits is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{different suits}) = \frac{\text{Number of ways to choose different suits}}{\text{Total ways to choose 2 cards}} = \frac{1014}{1326} \]
- Simplifying this fraction:
\[ P(\text{different suits}) = \frac{1014 \div 6}{1326 \div 6} = \frac{169}{221} \]

### Step 5: Finalize the answer
- The probability that the two cards drawn are of different suits is:
\[ \frac{169}{221} \]

### Conclusion
- The correct answer is not listed among the options provided, but the calculated probability is \(\frac{169}{221}\).