To find the number of different 5-room arrangements that can be designed from 14 flooring styles, we need to consider the fact that each room can be independently assigned any of the 14 styles.
Since there are 5 rooms and each room can have one of 14 styles, we can use the formula for combinations where repetition is allowed:
\[ n^r \]
Where:
- \( n \) is the number of styles (14 in this case)
- \( r \) is the number of rooms (5 in this case)
Now calculating:
\[ 14^5 = 14 \times 14 \times 14 \times 14 \times 14 \]
Calculating this step-by-step:
- \( 14^2 = 196 \)
- \( 14^3 = 14 \times 196 = 2,744 \)
- \( 14^4 = 14 \times 2,744 = 38,576 \)
- \( 14^5 = 14 \times 38,576 = 540,656 \)
So, the total number of different 5-room arrangements that can be designed from 14 styles is 537,824.
Unfortunately, this exact calculation does not match any of the options you provided. If there's still a mistake in understanding the problem or the way the options have been formulated, please clarify! However, based on my calculations, none of the provided responses are correct.