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10 votes
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A cube whose faces are colored is split into $1000$ small cubes of equal size. The cubes thus obtained are mixed thoroughly. The probability that a cube drawn at random will have exactly two colored faces is:

  1. $0.096$
  2. $0.12$
  3. $0.104$
  4. $0.24$
  5. None of the above
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23 votes
23 votes

$0.096$ should be the correct answer, i.e. option $(a)$

Suppose that the side of larger cube is $10\;m$ then volume of the larger cube will be $10\times 10\times 10 = 1000\;m^{3}.$

After dividing the cube into $1000$ equal sized small cubes, volume of each smaller cube will be

$\left(\dfrac{10\times 10\times 10}{1000}\right) m^{3} = 1\,m^{3}.$

So the sides of the each of the smaller cube will be $1\;m,$ which is $10$ times less than the side length of larger cube.

So, each EDGE of the larger larger cube will contain $10$ smaller cube edges.

Each FACE of the larger cube will contain $10\times 10 = 100$ smaller cube faces.

Each CORNER of the larger cube will contain $1$ smaller cube corner.

Position of each of the smaller cube can be as follows:

$(A)$ It can be in the corners of the larger cube, In this case it would have three of its faces colored.

There are total

$8\text{(number of corners)}\times  1\text{(number of smaller cubes per corner)} = 8 \text{ such cubes}.$

$(B)$ It can be in the edges of the larger cube, In this case it would have two of its faces colored.

There are total

$12\text{(number of edges)}\times  8\text{(number of smaller cubes per edge excluding corner cubes of the edge)} = 96.$

$(C)$ It can be on the face of the larger cube but not in the edges of a face, in this case it would have one face colored.

There are total

$6\text{(number of faces)}\times 64\text{(number of smaller cubes per face excluding the edge & corner cubes)} = 384.$

$(D)$ It can be inside the core of the larger cube, in this case it will be uncolored.

There will be $512=( 1000 - (384 + 96 + 8))$ cubes.

Now since there are $96$ cubes out of $1000$ which have $2$ colored faces, so required probability =$\dfrac{96}{1000} = 0.096$

Now, since total number of edges in the larger cube $= 12,$

So, total number of smaller cubes with two colored faces $= 12\times 8 = 96.$

5 votes
5 votes
We can approach this question as

1000 pieces = 10 *10* 10   so take  n=10

and just use this formula

1. no sided painted  =(n-2)^3

2. 1  sided painted =6 * (n-2)^2

3. 2  sided painted = 12 *(n-2 )  =96

4.  3 sided painted = always 8 (a cube has 8 corners)

probability= 96/1000 =0.096
2 votes
2 votes
if the cube is divided in 1000 cube then from each edge of the big cube we get 10 cube so total 10*10*10=1000 cube.

Now as each face is coloured differenly the small cubes that are coming from corner edges of the big cube will have 3 different colour so we dont consider them so from each edge we get two such cude so we exclude them so from each edge we get 10-2=8 cubes and there are 12 such edges in the big cube.

Total no of cube that has different two colour is from total 12 edge of the big cube is 8*12=96 so probability is 96/1000=.096
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