A rectangular floor measures 2 by 3 m. There are 5 white, 5 black, and 5 red parquet blocks available. Each block measures 1 by 1m. In how many different colors patterns can be floor be parqueted?

Question

Harsh Garg · Accepted Answer

The rectangular floor measures 2 by 3 meters, so it has 6 squares. There are 5 white, 5 black, and 5 red parquet blocks available, each measuring 1 by 1 meter.
We need to determine the number of different color patterns that can be used to parquet the floor.
Since there are 6 squares, we need to select 6 blocks from the 15 available (5 white, 5 black, and 5 red).
The number of ways to choose 6 blocks out of 15 is given by the multinomial coefficient:C(6; a, b, c) = 6! / (a! b! c!)where a, b, and c are the number of white, black, and red blocks, respectively, such that a + b + c = 6.
We need to sum this over all possible values of a, b, and c that sum to 6.
The number of different color patterns is the number of solutions to the equation:a + b + c = 6where 0 &le; a, b, c &le; 5.
This can be found using the stars and bars method:The number of non-negative integer solutions to the equation a + b + c = 6 is given by:C(6 + 2, 2) = C(8, 2) = 8! / (2! 6!) = 28
Thus, there are 28 different color patterns for parqueting the floor.