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(B) Let $n$ be the number of smaller cubes of edge length $1 \ cm$ obtained after cutting the bigger cube of edge length $5 \ cm$. Volume of bigger cu

Vedclass · Accepted Answer

$\frac{4}{49}$

Vedclass · Answer

(B) Let $n$ be the number of smaller cubes of edge length $1 \ cm$ obtained after cutting the bigger cube of edge length $5 \ cm$. Volume of bigger cube $= n 	imes$ Volume of smaller cube $\Rightarrow 5^3 = n 	imes 1^3$ $\Rightarrow n = 125$. When the painted cube is cut into $125$ small equal cubes: $1$. Number of cubes with $3$ faces painted (at corners) $= 8$. $2$. Number of cubes with $2$ faces painted (on edges) $= (5-2) 	imes 12 = 3 	imes 12 = 36$. $3$. Number of cubes with $1$ face painted (on faces) $= (5-2)^2 	imes 6 = 9 	imes 6 = 54$. Total number of cubes with at least one face painted $= 8 + 36 + 54 = 98$. We are given that the selected cube has at least one face painted. We need the probability that $2$ more faces are painted, which means the cube has $3$ faces painted in total. Let $E$ be the event that the cube has at least one face painted, so $n(E) = 98$. Let $F$ be the event that the cube has $3$ faces painted, so $n(F) = 8$. Since all cubes with $3$ faces painted are also cubes with at least one face painted, $F \subset E$. The required probability is $P(F|E) = \frac{n(F)}{n(E)} = \frac{8}{98} = \frac{4}{49}$.

$A$ cube having an edge of length $5 \ cm$ is painted on all faces and then it is cut into equal cubes of unit volume. $A$ small cube is selected at random and it is found that at least one of its faces is painted. What is the probability that $2$ more faces of it are also painted?

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