Respuesta :
1260 different towers can be made with a height of 8 cubes with 2 red cubes, 3 blue cubes and 4 green cubes
There are three cases to consider, each one where a cube of either red, blue or green is left out.
Case 1:
A red cube is left out so the tower will comprise of 4 green cubes, 3 blue cubes and 1 red cube.
Let’s consider the green cubes first. There are 8 possible locations available for the first green cube, leaving 7 possible locations for the second green cube, leaving 6 locations for the third cube and 5 locations for the fourth cube.
Thus, the green cubes can be allocated in 8x7x6x5 ways. However for each particular allocation there are identical alternatives in which the same locations are allocated but the cubes are placed in them in a different sequence.
Thus there are (8x7x6x5)/24 ways that green cubes can be distributed within the tower.
Since the green cubes have been allocated this leaves only four locations to place the three blue cubes, so these can be allocated in 4x3x2 ways, and of these 3! (=6) are identical. So there are (4x3x2)/6 ways that the blue cubes can be distributed within the tower.
If the green and blue cubes have been allocated then there’s only 1 way to distribute the red cube.
Thus, the number of ways the cubes in case 1 can be distributed is:
(8x7x6x5)/24 x (4x3x2)/6 x 1
= 1680/24 x 24/6 x 1
= 70 x 4 x 1 = 280 different ways.
It is important to note that the order we consider the colors of the cubes is unimportant as the same result would be obtained.
For example, considering the cubes in the order blue, red then green would give the calculation:
(8x7x6)/6 x (5)/1 x (4x3x2x1)/24
= 336/6 x 5/1 x 24/24
= 56 x 5 x 1 = 280]
Case 2:
A blue cube is left out so the tower will comprise of 4 green cubes, 2 blue cubes and 2 red cubes.
By the same logic as applied in case 1, the calculation of the number ways the cubes can be distributed is:
(8x7x6x5)/24 x (4x3)/2 x (2x1)/2
= 1680/24 x 12/2 x 1
= 70 x 6 x 1 = 420 different ways.
Case 3:
A green cube is left out so the tower will comprise of 3 green cubes, 3 blue cubes and 2 red cubes.
Following the same logic again:
(8x7x6)/6 x (5x4x3)/6 x (2x1)/2
= 336/6 x 60/6 x 2/2
= 56 x 10 x 1 = 560 different ways.
Finally, we have enumerated the unique distributions of colored cubes in the three possible cases. The total number of different towers is the sum of different ways the cubes can be distributed in all three cases .
So, 280 + 420 + 560 = 1260 different towers.
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