Box A has 3 white and 2 red balls, box B has 2 white and 4 red balls. If two balls are selected at random without replacements from the box A and 2 more are selected at random from B, the probability that all the four balls are white is

Options

A. `1/10`
B. `1/50`
C. `1-((.^(2)C_(2))/(.^(5)C_(2)).(.^(4)C_(2))/(.^(5)C_(2)))`
D. `(.^(3)C_(2).^(2)C_(2))/(.^(5)C_(2).^(6)C_(2))`

Answer

To solve the problem, we need to find the probability that all four balls selected (two from Box A and two from Box B) are white. Let's break down the solution step by step.

### Step 1: Identify the contents of the boxes
- **Box A** contains 3 white balls and 2 red balls.
- **Box B** contains 2 white balls and 4 red balls.

### Step 2: Calculate the total number of ways to choose 2 balls from Box A
The total number of balls in Box A is 5 (3 white + 2 red). The number of ways to choose 2 balls from Box A is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose.

\[ \text{Total ways to choose 2 balls from Box A} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Calculate the number of ways to choose 2 white balls from Box A
We want to choose 2 white balls from the 3 available white balls in Box A.

\[ \text{Ways to choose 2 white balls from Box A} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

### Step 4: Calculate the total number of ways to choose 2 balls from Box B
The total number of balls in Box B is 6 (2 white + 4 red). The number of ways to choose 2 balls from Box B is:

\[ \text{Total ways to choose 2 balls from Box B} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Step 5: Calculate the number of ways to choose 2 white balls from Box B
We want to choose 2 white balls from the 2 available white balls in Box B.

\[ \text{Ways to choose 2 white balls from Box B} = \binom{2}{2} = \frac{2!}{2!(2-2)!} = 1 \]

### Step 6: Calculate the probability of selecting all white balls
Now, we can calculate the probability of selecting 2 white balls from Box A and 2 white balls from Box B.

The probability \( P(D) \) that all four balls are white is given by:

\[ P(D) = \frac{\text{Ways to choose 2 white from A} \times \text{Ways to choose 2 white from B}}{\text{Total ways to choose 2 from A} \times \text{Total ways to choose 2 from B}} \]

Substituting the values we calculated:

\[ P(D) = \frac{3 \times 1}{10 \times 15} = \frac{3}{150} = \frac{1}{50} \]

### Final Answer
The probability that all four balls selected are white is \( \frac{1}{50} \).

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