How many numbers from 100 000 to 112 000 contain at least one zero?

Answer 1

Well every number between 100,000 and 109,999 will have a zero in the 10 thousands place, so that is 10,000 right off the bat.

Similarly the numbers 110,000 to 110,999 all have a zero in the thousands place, so that is another 1,000.

That leaves the numbers between 111,000 and 112,000.

You could count them...
111,000 to 111,099 --> 100
111,100 to 111,109 --> 10
111,110 to 111,199 --> 9
111,200 to 111,209 --> 10
111,210 to 111,299 --> 9
...
111,900 to 111,909 --> 10
111,910 to 111,999 --> 9
112,000 --> 1
---------------
Total in this range: 100 + 9(19) + 1 = 272

But an easier way might be to use combinations. You have the last 3 digits. How many combinations result in a number where at least 1 of these digits is zero?

Well, the easier question is what is the chance that *none* of the digits are zero. That would be 9 choices for the first digit, 9 choices for the second digit and 9 choices for the last digit.
9 x 9 x 9 = 729. So 729 combinations have *no* zero. Since 111,000 to 112,000 is 1001 numbers, that leaves 272 numbers that must have at least one zero, same as what we counted manually.

Either way, the total would be 11,272, if you are inclusive of 100,000 and 112,000.

(Note: If you mean exclusively *between* 100,000 and 112,000, the answer is two less, or 11,270.)

Answer 2

Sometimes it helps to turn the question around and ask its inverse, such as in this instance of how many numbers between 100,000 and 112,000 lack zeroes as digits. In this case, you will find a total of 12,001 numbers between 100,000 and 112,000, including both end numbers.

Because we are now counting numbers that lack zeroes, the top number, 112,000, is not included in that count. Of the 12,000 numbers between 100,000 and 111,999 inclusive, all lack a zero in the hundred thousands digit and exactly ten thousand have a zero in the ten-thousands digit; the remaining 2,000 have a 1 in this digit.

Of the 2,000 numbers between 110,000 and 111,999, the top half, or the top 1,000, lack a zero in the thousands digit.

Of the top 1,000, nine of every ten will lack a zero in the hundreds digit.

Of the 900 remaining, nine of ten also lack a zero in the tens digit.

Of the 810 remaining, nine of ten, or 729, lack a zero in the ones digit. These 729 have no place left for a zero and have no zeroes at all. Subtracting this 729 from the original 12,001 numbers, gives 11,272 numbers with at least one zero. Both end numbers have zeroes, so you can subtract one or two from this total if you want to exclude one or both end numbers from the count.

Answer 3

wow.
In this a MENSA test question....?