Thuy rolls a number cube 7 times. which expression represents the probability of rolling a 4 exactly 2 times?

The result of rolling a number cube 7 times is a 7-digit number composed of digits 1,2,3,4,5 and 6 so that digits can repeat. The total number of possibilities is 6^7.

The number of possibilities where 4 appears exactly two times is 5^5 * (7!-6!/2).

5^5 is the number of 5-digits numbers composed of digits 1,2,3,5 and 6 so that digits can repeat.

7! is the number of permutations of digits 1,2,3,4,4,5 and 6.

6! is the number of permutations of digits 1,2,3,{4,4},5 and 6.

We don't want to subtract all numbers where digits 4 appear side by side. That's why we must divide 6! by 2.

Finally, the probability is P=5^5 (7!-6!/2) / 7^7