Three cards are drawn randomly from a pack of 52 cards , then find the probability that 2 cards being red cards & 1 card being black card.

Options

A. `(13)/(34)`
B. `(19)/(52)`
C. `(7)/(39)`
D. `(3)/(43)`

Answer

To find the probability of drawing 2 red cards and 1 black card from a pack of 52 cards, we can follow these steps:

### Step 1: Determine the total number of ways to draw 3 cards from 52 cards.
The total number of ways to choose 3 cards from 52 is given by the combination formula:
\[ \text{Total ways} = \binom{52}{3} \]

### Step 2: Determine the number of ways to draw 2 red cards from 26 red cards.
The number of ways to choose 2 red cards from 26 red cards is:
\[ \text{Ways to choose 2 red cards} = \binom{26}{2} \]

### Step 3: Determine the number of ways to draw 1 black card from 26 black cards.
The number of ways to choose 1 black card from 26 black cards is:
\[ \text{Ways to choose 1 black card} = \binom{26}{1} \]

### Step 4: Calculate the total number of favorable outcomes.
The total number of favorable outcomes (2 red cards and 1 black card) can be calculated by multiplying the number of ways to choose red cards and black cards:
\[ \text{Favorable outcomes} = \binom{26}{2} \times \binom{26}{1} \]

### Step 5: Calculate the probability.
The probability of drawing 2 red cards and 1 black card is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{\binom{26}{2} \times \binom{26}{1}}{\binom{52}{3}} \]

### Step 6: Substitute the values and simplify.
Now we can calculate the combinations:
- \(\binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325\)
- \(\binom{26}{1} = 26\)
- \(\binom{52}{3} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22100\)

Now substituting these values into the probability formula:
\[ \text{Probability} = \frac{325 \times 26}{22100} = \frac{8450}{22100} \]

### Step 7: Simplify the fraction.
To simplify \(\frac{8450}{22100}\), we can divide both the numerator and the denominator by 50:
\[ \frac{8450 \div 50}{22100 \div 50} = \frac{169}{442} \]

Thus, the probability that 2 cards are red and 1 card is black is:
\[ \frac{169}{442} \]