Problem Solving

The question is from the 300 GMAT problems with best solutions from our forum.

A password of a computer used five digits where they are from 0 and 9. What is the probability that the password solely consists of prime numbers and zero?
A 1/32
B 1/16
C 1/8
D 2/5
E ½

My approach:

We often use permutation and combinations in probability to arrive at no of outcomes.

Total number of outcomes = 10C5 - Choosing 5 digits out of 10 digits.
Total number of favorable outcomes = 5C5 x 5! ((Choosing 5 digits - 4 prime and zero out of the same) x (the number of ways each digit can be arranged))

5C5 = 1
so P(Password being prime and zero) = (1 x 5!)/10C5

The solution given in the document is different from the way i solved it. There are many problems which i have solved it using combinations and this one is confusing as even the answer choices won't fit in if i solve it using the above approach.

Need your help on why the above approach is wrong.


OA is A

Solution given in the document:
There are 10 possible options (0,1,2,3,4,5,6,7,8,9) for each digit.
5 of the options (0,2,3,5,7) are zero or prime.
So, P(a given digit is zero or prime) = 5/10 = 1/2

A quick way is to look at this as an AND probability.
P(all five digits are zero or prime) = P(1st digit is zero or prime AND 2nd digit is zero or prime AND 3rd digit is zero or prime AND 4th digit is zero or prime AND 5th digit is zero or prime)

This is equal to P(1st digit is zero or prime) x P(2nd digit is zero or prime) x P(3rd digit is zero or prime) x P(4th digit is zero or prime) x P(5th digit is zero or prime)
So, we get 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32

Thanks in Advance.