So,

Total ways = 6*6*6

WITH restrictions
Ways to choose 2 with same number out of 23 = 3C2=3 ways..
These 2 can have any of the 6 sides = 6 ways
The third can take any of the remaining 5, so 5 ways
Total = 3*5*6

Probability = \(\frac{3*5*6}{6*6*6}=\frac{5}{12}\)

We can also use the complement here. The total number of possibilities is 6*6*6 = 216.

In 6 of these possibilities, all the dice are the same number. That's a probability of \(\frac{6}{216}\).

What's the probability that none of the dice come up as the same number?

Well, we have a probability of 1 that the first die will be a number. Then we have a \(\frac{5}{6}\) probability that the next die will not match it. And a \(\frac{4}{6}\) probability that the third will not match either of the previous two. Together: \(1 \times \frac{5}{6} \times \frac{4}{6} = \frac{20}{36} = \frac{120}{216}\).

So we have the probability that all the dice are the same, and that none are the same. The only other remaining possibility is that two dice are the same. So:

\(1 - \frac{6}{216} - \frac{120}{216} = \frac{90}{216} = \frac{5}{12}\)

Given that Three dice are rolled simultaneously and We need to find What is the probability that exactly two of the dice will come up as the same number?

As we are rolling three dice => Number of cases = \(6^3\) = 216

Out of the three rolls lets find out the two rolls in which we will get the same number.
We can get that in 3C2 ways = \(\frac{3!}{2!*1!}\) = 3 ways

Now these two numbers can be any number out of 6 => First number we can pick in 6 ways and second number we can pick in 1 way as it has to be the same as the first number

For the third number we have 5 ways as it has to be different from the first two

=> Total number of ways = 3 * 6 * 1 * 5

=> Probability that exactly two of the dice will come up as the same number = \(\frac{3 * 6 * 1 * 5}{6^3}\) = \(\frac{5}{12}\)

So, Answer will be A
Hope it helps!

Watch the following video to learn How to Solve Dice Rolling Probability Problems

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