A card is drawn from a pack of 52 playing cards. The card is replaced and the pack is reshuffled. If this is done six times. The probability that 2 hearts, 2 diamond and 2 black cards are drawn is

Options

A. `90xx((1)/(4))^6`
B. `(45)/(2)xx((3)/(4))^4`
C. `90xx((1)/(2))^10`
D. none of these

Explanation

To solve the problem of finding the probability of drawing 2 hearts, 2 diamonds, and 2 black cards from a pack of 52 playing cards over 6 draws (with replacement), we can follow these steps:

### Step 1: Determine the probabilities of drawing each type of card.
- **Probability of drawing a heart (P(H))**: There are 13 hearts in a deck of 52 cards.
\[ P(H) = \frac{13}{52} = \frac{1}{4} \]

- **Probability of drawing a diamond (P(D))**: Similarly, there are 13 diamonds.
\[ P(D) = \frac{13}{52} = \frac{1}{4} \]

- **Probability of drawing a black card (P(B))**: There are 26 black cards (13 spades and 13 clubs).
\[ P(B) = \frac{26}{52} = \frac{1}{2} \]

### Step 2: Calculate the total number of ways to arrange the draws.
We need to find the number of ways to arrange 2 hearts, 2 diamonds, and 2 black cards in 6 draws. This can be calculated using the multinomial coefficient:
\[ \text{Number of arrangements} = \frac{6!}{2! \times 2! \times 2!} \]
Calculating this gives:
\[ 6! = 720 \quad \text{and} \quad 2! = 2 \]
Thus,
\[ \text{Number of arrangements} = \frac{720}{2 \times 2 \times 2} = \frac{720}{8} = 90 \]

### Step 3: Calculate the probability of drawing 2 hearts, 2 diamonds, and 2 black cards.
The probability of drawing 2 hearts, 2 diamonds, and 2 black cards in any specific arrangement is:
\[ P(H)^2 \times P(D)^2 \times P(B)^2 = \left(\frac{1}{4}\right)^2 \times \left(\frac{1}{4}\right)^2 \times \left(\frac{1}{2}\right)^2 \]
Calculating this:
\[ = \frac{1}{16} \times \frac{1}{16} \times \frac{1}{4} = \frac{1}{256} \times \frac{1}{4} = \frac{1}{1024} \]

### Step 4: Combine the number of arrangements and the probability.
The total probability of drawing 2 hearts, 2 diamonds, and 2 black cards is:
\[ \text{Total Probability} = \text{Number of arrangements} \times \text{Probability of a specific arrangement} \]
\[ = 90 \times \frac{1}{1024} = \frac{90}{1024} = \frac{45}{512} \]

### Final Answer:
The probability that 2 hearts, 2 diamonds, and 2 black cards are drawn is:
\[ \frac{45}{512} \]

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