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 picking 7 shoes from 12 pairs such that there are exactly 2 complete pairs, we can break down the solution into several steps:

### Step 1: Choose 2 pairs from 12 pairs
We need to select 2 pairs of shoes from the 12 available pairs. The number of ways to choose 2 pairs from 12 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 pairs} = \binom{12}{2} \]

Calculating this:

\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

### Step 2: Choose 3 additional shoes from the remaining pairs
After selecting 2 pairs, we have 10 pairs left. We need to select 3 more shoes, but we want to ensure that these 3 shoes do not form another complete pair. This means we can only select one shoe from each of the 3 different pairs.

The number of ways to choose 3 pairs from the remaining 10 pairs is:

\[ \text{Number of ways to choose 3 pairs} = \binom{10}{3} \]

Calculating this:

\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 3: Choose one shoe from each of the selected pairs
For each of the 3 pairs chosen in the previous step, we can choose either the left shoe or the right shoe. Since there are 3 pairs, the number of ways to choose one shoe from each of these pairs is:

\[ \text{Ways to choose one shoe from each pair} = 2^3 = 8 \]

### Step 4: Calculate the total number of ways
Now, we can find the total number of ways to pick the shoes by multiplying the number of ways to choose the pairs and the number of ways to choose the shoes from those pairs:

\[ \text{Total ways} = \left(\text{Ways to choose 2 pairs}\right) \times \left(\text{Ways to choose 3 pairs}\right) \times \left(\text{Ways to choose shoes from those pairs}\right) \]

Substituting the values we calculated:

\[ \text{Total ways} = 66 \times 120 \times 8 \]

Calculating this:

\[ 66 \times 120 = 7920 \]

Then,

\[ 7920 \times 8 = 63360 \]

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

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