Four friends have 7 shirts, 6 paints and 8 ties. In how many ways can they wear them?

Options

A. A. 336000
B. B. 33600
C. C. 12480
D. D. 508032000

Answer

To solve the problem of how many ways four friends can wear 7 shirts, 6 pants, and 8 ties, we will use the concept of permutations.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 4 friends and they can choose from 7 shirts, 6 pants, and 8 ties. We need to find out how many different combinations of outfits they can wear.

2. **Calculating the Permutations for Shirts**:
- The number of ways to choose and arrange 4 shirts from 7 shirts is given by the permutation formula \( nPr = \frac{n!}{(n-r)!} \).
- Here, \( n = 7 \) and \( r = 4 \).
- So, \( 7P4 = \frac{7!}{(7-4)!} = \frac{7!}{3!} = \frac{7 \times 6 \times 5 \times 4}{1} = 840 \).

3. **Calculating the Permutations for Pants**:
- The number of ways to choose and arrange 4 pants from 6 pants is similarly calculated.
- Here, \( n = 6 \) and \( r = 4 \).
- So, \( 6P4 = \frac{6!}{(6-4)!} = \frac{6!}{2!} = \frac{6 \times 5 \times 4 \times 3}{1} = 360 \).

4. **Calculating the Permutations for Ties**:
- The number of ways to choose and arrange 4 ties from 8 ties is calculated in the same manner.
- Here, \( n = 8 \) and \( r = 4 \).
- So, \( 8P4 = \frac{8!}{(8-4)!} = \frac{8!}{4!} = \frac{8 \times 7 \times 6 \times 5}{1} = 1680 \).

5. **Calculating the Total Combinations**:
- Since the choices of shirts, pants, and ties are independent, we multiply the number of ways to choose each item.
- Therefore, the total number of ways the friends can wear the outfits is:
\[ \text{Total Ways} = 7P4 \times 6P4 \times 8P4 = 840 \times 360 \times 1680 \]

6. **Final Calculation**:
- Now we calculate \( 840 \times 360 \times 1680 \).
- First, calculate \( 840 \times 360 = 302400 \).
- Then, calculate \( 302400 \times 1680 = 508032000 \).

Thus, the total number of ways the four friends can wear the shirts, pants, and ties is **508032000**.