To determine how many different 5-room arrangements can be designed from 14 styles of flooring, we can use the combinatorial concept of combinations with repetition, since each room can have any of the 14 styles independently of the others.
Here we use the formula for combinations with repetition, which is given by:
\[ C(n+r-1, r) \]
where \( n \) is the number of styles (14), \( r \) is the number of rooms (5), and \( C \) denotes a combination.
Plugging in the values:
\[ C(14 + 5 - 1, 5) = C(18, 5) \]
Now we calculate \( C(18, 5) \):
\[ C(18, 5) = \frac{18!}{5!(18-5)!} = \frac{18!}{5! \cdot 13!} \]
Calculating the factorials:
\[ C(18, 5) = \frac{18 \times 17 \times 16 \times 15 \times 14}{5 \times 4 \times 3 \times 2 \times 1} \]
Calculating the numerator:
\[ 18 \times 17 = 306 \]
\[ 306 \times 16 = 4896 \]
\[ 4896 \times 15 = 73440 \]
\[ 73440 \times 14 = 1028160 \]
Now we calculate the denominator:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Now divide the numerator by the denominator:
\[ C(18, 5) = \frac{1028160}{120} = 8568 \]
Thus, the total number of different 5-room arrangements from 14 styles is \( 8568 \).
None of the provided responses match this calculation, suggesting there may be a misunderstanding or misinterpretation in the original question or options. The correct answer based on combinations with repetition, however, is \( 8568 \).