There are `12` pairs of shoes in a box. Then the possible number of ways of picking `7` shoes so that there are exactly two pairs of shoes are

Options

A. `63360`
B. `63300`
C. `63260`
D. `63060`

Explanation

To solve the problem of finding the number of ways to pick 7 shoes from 12 pairs such that there are exactly 2 complete pairs and 3 individual shoes, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify Total Shoes**:
- There are 12 pairs of shoes, which means there are a total of \( 12 \times 2 = 24 \) shoes.

2. **Choose 2 Pairs**:
- We need to choose 2 pairs of shoes from the 12 pairs. The number of ways to choose 2 pairs from 12 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose 2 pairs} = \binom{12}{2} \]

3. **Choose 3 Additional Shoes**:
- After selecting 2 pairs (which gives us 4 shoes), we need to select 3 more shoes from the remaining pairs. Since we have already chosen 2 pairs, we have \( 12 - 2 = 10 \) pairs left. We can choose any 3 pairs from these 10 pairs:
\[ \text{Ways to choose 3 pairs} = \binom{10}{3} \]

4. **Select Shoes from Chosen Pairs**:
- From the 3 pairs we selected, we need to choose 1 shoe from each pair. For each pair, we have 2 choices (left shoe or right shoe). Therefore, for 3 pairs, the number of ways to choose the shoes is:
\[ \text{Ways to choose shoes from 3 pairs} = 2^3 = 8 \]

5. **Calculate Total Ways**:
- Now, we can find the total number of ways to select the shoes by multiplying the number of ways to choose the pairs and the number of ways to select the shoes from those pairs:
\[ \text{Total ways} = \binom{12}{2} \times \binom{10}{3} \times 2^3 \]

6. **Calculate Each Component**:
- Calculate \( \binom{12}{2} \):
\[ \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66 \]
- Calculate \( \binom{10}{3} \):
\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
- The total number of ways is:
\[ \text{Total ways} = 66 \times 120 \times 8 \]

7. **Final Calculation**:
- Calculate \( 66 \times 120 \):
\[ 66 \times 120 = 7920 \]
- Now multiply by 8:
\[ 7920 \times 8 = 63360 \]

### Final Answer:
The total number of ways to pick 7 shoes such that there are exactly 2 pairs of shoes is **63360**.