[1000] How many numbers can you come up with using the numbers 1-7 for a 7-digit number?

We have a 7-digit number to fill, so we have one slot for each digit:

How many numbers can we choose from for the first slot? Well, we can
choose 7.

How many numbers can we choose for the second slot? We can choose 7
again.

And the third slot? 7 again.

I'm sure the rocket scientists among you have figured out that you can
choose 7 numbers for the fourth, fifth, sixth and seventh slots as
well.

So, the final answer is 7x7x7x7x7x7x7 =
823,543

[1001] How many numbers can you come up with using the numbers 1-7 for a 7 digit number if you use the numbers only once?

We have a 7-digit number to fill, so we have one slot for each digit:

How many numbers can we choose from for the first slot? Well, we can
choose 7.

How many can we choose for the second slot? Well, since we already used
a number for the first slot, we only have 6 to choose from now.

For the third slot, we have 5 numbers.

For the fourth slot, we have 4 numbers.

For the fifth slot, we have 3 numbers.

For the sixth slot, we have 2 numbers.

And, of course, that leaves 1 number for the last slot.

So, the final answer is 7x6x5x4x3x2x1 = 7! = 5040

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