Question

A cube painted red on all sides is cut into 125 equal small cubes. A small cube when picked up is found to show red colour on one side of its face. The probability that two more faces also show red colour is

• A. $\frac{4}{49}$
• B. $\frac{1}{13}$
• C. $\frac{3}{49}$
• D. $\frac{4}{51}$

Answer 1

The correct option is A $\frac{4}{49}$

To cut it into $125$ congruent smaller cubes required $5$ horizontal cuts and $5$ vertical cuts. The only cubes that are pointed red on one side are the interior cubes of each side of $25$ cubes sides, $9$ of them are painted red on only one side.

i.e, $9\times 6=54$ cubes that are not painted at all are the interior cubes, there are $3\times 3\times 3=27$ interior cubes.

Of all $125$ cubes,

$27$ are interior cubes, that have no sides painted red.

$54$ are cubes painted on only one side.

$36$ are cubes that are painted red on two side.

$8$ are corner cubes, painted red on $3$ sides.

Probability that $3$ faces shows red color

$\Rightarrow$ The probability of red side facing up $=\frac{27}{125}\times \frac{0}{6}+\frac{54}{125}\times \frac{1}{6}+\frac{36}{125}\times \frac{2}{6}+\frac{8}{125}\times \frac{3}{6}=\frac{1}{6}$

$\Rightarrow$ Probability of two more red colour facing up $=\frac{1}{6}\times \frac{4\times 6}{49}=\frac{4}{49}.$

Hence, the answer is $\frac{4}{49}.$