PS on Probability

[satishchandra]

A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. If each block measures 1 by 1 meter, in how many different color patterns can the floor be parqueted?

(A) 104
(B) 213
(C) 577
(D) 705
(E) 726

[shankar.ashwin]

A 2X3 floor requires 6 (1X1) square tiles - Atleast 2 colors

Each tile could be of any of the 3 colors, so we have 3^6 possibilities (for 6 tiles), but we have only 5 tiles of the same color

So subtract 3 cases which included all white/black/red

So, 3^6 - 3 = 729-3 = 726

[Anurag@Gurome]

A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. If each block measures 1 by 1 meter, in how many different color patterns can the floor be parqueted?

(A) 104
(B) 213
(C) 577
(D) 705
(E) 726

The rectangular floor measures 2 by 3 meters.
Thus there are 2*3 = 6 blocks of measurement 1 by 1 meter.

Now there are 3 possible color for each block.
Thus if we had infinite numbers of parquet blocks of each color, we would've done the decoration in 3^6 = 729 ways.

But we have a limited number of parquet blocks of each color, i.e 5 of each. Therefore all of the blocks cannot be of the same color at the same time. Thus all of the 6 blocks are white or black or red is not possible. Therefore except these 3 impossible cases the scenario is same as if we have infinite numbers of parquet blocks.

Therefore, actual number of different patterns = (729 - 3) = 726

The correct answer is E.