Explanation

This problem relies on the fundamental counting principle, which says that the total number of ways for something to happen is the product of the number of options for each individual choice.

The problem asks how many five-digit numbers can be created from the digits 5, 6, 7, 8, 9, and 0. For the first digit, there are only five options (5, 6, 7, 8, and 9) because a five-digit number must start with a non-zero integer.

For the second digit, there are 5 choices again, because now zero can be used but one of the other numbers has already been used, and numbers cannot be repeated. For the third
number, there are 4 choices, for the fourth number there are 3 choices, and for the fifth number there are 2 choices. Thus, the total number of choices is (5)(5)(4)(3)(2) = 600.

Alternatively, use the same logic and realize there are 5 choices for the first digit. (Separate out the first step because you have to remove the zero from consideration.) The remaining five digits all have an equal chance of being chosen, so choose four out of the remaining five digits to complete the number. The number of ways in which this second step can be accomplished is \(\frac{5!}{1!}= (5)(4)(3)(2)\).

Thus, the total number of choices is again equal to (5)(5)(4)(3)(2) = 600.

Lets make this easier.
we know that the total number of combinations is 6!
but we are limited by not being able to put a 0 as the beginning digit.

we know then that we have 6! - (all combinations with 0 as the beginning digit)

to find out all the combinations where the number begins with 0, is just 5!.

just subtract 6! - 5! and you have 720-120=600.. D is the answer.

in case you get confused how to calculuate all combinations with 0 as the beginning digit, keep reading...

Just think

line up the possible number combinations

we know that zero is the beginning digit, so there is only one combination for that

1

but we have 5 number choices remaining for the next digit
1*5
we have 4 number choices remaining for the next digit.
1*5*4

and so on and so on until we have:
1*5*4*3*2*1=120

hope that helped!

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