How many combinations can be made out of five digits?

Answer 1

Let's try and figure this one out...

First case (the easy one) - We want to use any of the 10 digits to create a 5-digit number. In this case, we have 5 slots and in each slot we can put any of 10 digits.

So the answer is 10*10*10*10*10 = 510 = 100,000 combinations


Second case - We want to create a 5-digit number using any of the 10 digits, but we only want to use each digit once. This turns out to be only a little more difficult.

Let's look at each digit individually.
First digit - We can use any digit from [0-9], so we have 10 possibilities.
Second digit - We can use any digit from [0-9] except for the one used in the first digit, so we now have 9 possibilities.
Third digit - We can use any digit from [0-9] except for the ones used in the first and second digits, so we now have 8 possibilities.
Fourth digit - As above, except we now have 7 possible choices.
Fifth digit - As above, except we now have 6 possible choices.

So our final number of combinations is 10 * 9 * 8 * 7 * 6 = 30,240 combinations


Special case - What happens if we want to create a number which does not begin with a zero? Well, we can make a simple adjustment to either of the above cases to take care of this. Just observe that the first digit is not limited to 9 possibilities [1-9], not 10.

Special first case = 9 * 10 * 10 * 10 * 10 = 90,000 combinations
Special second case = 9 * 9 * 8 * 7 * 6 = 27,216 combinations