Question

Probaility Question?

Answer 1

See below:

Explanation:

Let's first talk about the number of credit card numbers we can make (this ignores all the numerical codes within the card number itself, referring to issuing agency, etc).

There are 16 places and each place can take any one of ten digits and so we can make:

`10^16=10,000,000,000,000,000` card numbers we can make.

In our question, we have a set of digits and are asked for the number of ways we can order the digits. If all the digits were unique, there'd be `16! ~~ 2.1xx10^13` ways to order the digits. However, we have duplicate numbers, so we need to eliminate the duplicates. There are:

2 4s
3 2s
3 3s
2 0s

To eliminate the duplicates, we divide by the internal ordering of each group. For instance, there is `3! = 6` ways to order the 2s. In fact, the internal ordering for each group is the number of items in that group, then taken to the factorial. And so we have:

`(16!)/(2!3!3!2!)=(16!)/(2xx6xx6xx2)=145,297,152,000` ways to order the numbers.

The probability that someone can decrypt the card number knowing all the digits is:

`1/(145,297,152,000)~~7xx10^(-12)`

If someone didn't know the numbers and tried to guess all 16 digits correctly, the probability would be:

`1/(10^16)=1xx10^(-16)`