Number of ways of selecting 4 shoes from 10 pairs such that exactly one correct pair is selected is (A) ` .^(10)C_(1) .^(9)C_(2).2^(2)` (B) ` .^(10)C_(1)xx9xx16` (C) . `(( .^(10)C_(1).^(4)P_(2).18xx16)/(4!))` (D) d. `10xx18xx16`

Options

A. `a. .^(10)C_(1) .^(9)C_(2).2^(3)`
B. b. ` .^(10)C_(1)xx9xx16`
C. c. `(( .^(10)C_(1).^(4)P_(2).18xx16)/(4!))`
D. d. `10xx18xx16`

Explanation

To solve the problem of selecting 4 shoes from 10 pairs such that exactly one correct pair is selected, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Select the Correct Pair:**
We need to select exactly one correct pair of shoes from the 10 pairs available. The number of ways to choose 1 correct pair from 10 pairs is given by:
\[ \binom{10}{1} \]

2. **Select Remaining Shoes:**
After selecting one correct pair, we have 9 pairs left. From these 9 pairs, we need to select 2 shoes such that they do not form a correct pair. This means we need to select shoes from 2 different pairs.

3. **Choosing 2 Pairs from 9:**
The number of ways to choose 2 pairs from the remaining 9 pairs is given by:
\[ \binom{9}{2} \]

4. **Selecting Shoes from the Chosen Pairs:**
For each of the 2 pairs selected, we can choose either the left shoe or the right shoe. Therefore, for each of the 2 pairs, we have 2 choices (left or right). Thus, the total number of ways to select the shoes from these 2 pairs is:
\[ 2 \times 2 = 2^2 \]

5. **Combining All Choices:**
Now, we can combine all the choices made in the previous steps to find the total number of ways to select the shoes. The total number of ways is given by:
\[ \text{Total Ways} = \binom{10}{1} \times \binom{9}{2} \times 2^2 \]

### Final Calculation:
Putting it all together, we have:
\[ \text{Total Ways} = \binom{10}{1} \times \binom{9}{2} \times 2^2 \]

### Conclusion:
Thus, the number of ways of selecting 4 shoes from 10 pairs such that exactly one correct pair is selected is:
\[ \binom{10}{1} \times \binom{9}{2} \times 2^2 \]
This matches option (A) from the given choices.