A number plate has 6 single digit numbers If a plate is made

A number plate has 6 single digit numbers. If a plate is made by selecting digits from 0 to 9 randomly, what is the probability that there will be exactly two 9s and two 0s?

Solution

There are 10^6 such plates.

If we set the 4 numbers to be 9, 9, 0, 0, then we have 8*7 = 56 to choose the other two numbers, if they are distinct.

Then, there are 6!/(2!2!1!1!) = 180 ways to permute those. Thus, there are 56*180 = 10080 such plates.

If the last two numbers are the same, there are 8 ways to do it. Then there are 6!/(2!2!2!) = 90 ways to permute that. Thus, there are 8*90 = 720 ways to do it.

Thus, all in all, there are 10080 + 720 = 10800 ways.

Thus, the probability is

P = 10800 / 1000000 = 0.0108 [ANSWER]