All the possible five-digit numbers are formed using only the digits

how is it 3 raise to the power 4 ?

Five digit number: ____ ____ ____ ____ ____

Each blank can be filled up in 3 ways. So there are 3 * 3 * 3 * 3 * 3 = 243 ways of making the five digit number.

In how many of them will there be no 5? There will be no 5 in numbers made from only 4 and 6. So each spot will have only 2 options in that case.
Then there are 2 * 2 * 2 * 2 * 2 = 32 ways of making the five digit number.

In how many of them will there be only one 5? Select 1 spot for the 5 out of 5 spots in 5 ways.
The number of ways of filling up each of the other 4 spots is 2.
Total numbers = 5 * 2 * 2 * 2 * 2 = 80 ways of making the five digit number.

So in 32 + 80 = 112 numbers, there will be no 5 or only one 5.

In the other 243 - 112 = 131 numbers, there will be at least two 5s.

Answer (A)

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