5 digit numbers can be selected from the numbers {0,2,4,5,6}

Question

How many different 5 digit numbers can be selected from the numbers {0,2,4,5,6} (Generally 01 and 1 are considered as a one number.)

Is the answer 5! ?

Answer 1

If $02456$ is a 4 digit number then the answer is not $5!$. This is because the number cannot start with a $0$ and 4 remaining digits if it's a 5 digit number. There are two approaches to answer this. Think of each slot for the digits. For the first place, you have a choice of 4 digits(because 0 is not allowed), for the second place, you still have four choices(the 3 remaining non zero digits and the 0 digit), for the third place you have 3 choices(the 2 remaining non zero digits and the 0 digits) and so on. This yields an answer $4 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 96$

Another way to approach this is by subtracting all the permutations that start with a $0$ from all the possible permutations. There are $5!$ permutations without restrictions. If the number starts with zero, there are only $4$ choices left for the remaining digits meaning there are $4!$ permutations with a $0$ at the front. Subtraction yields $5! - 4! = 120 - 24 = 96$ which is the same answer from the method above.