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How many three-digit integers greater than 710 are there such that all their digits are different?

A. 198 B. 202 C. 207 D. 209 E. 212

First find how many integers between 700 and 999 are such that all their digits are different.

We have : \(\text{(3 options for the first digit)}*\text{(9 options for the second digit)}*\text{(8 options for the third digit)} = 216\) numbers.

Among these 216 numbers, 9 (701, 702, 703, 704, 705, 706, 708, 709, 710) are not bigger than 710. The answer to the question is therefore \(216 - 9 = 207\).

Hi, 216 is the total number where all digits are different.. 707 has 7 at two places, so 707 has not been taken as a part of these 216 numbers..
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Hi guys, I understand the solution above but have trouble seeing what I did wrong - can you please help?

If the first digit is 7: 2nd slot can be anything from #2 to #9 with the exception of 7, so that's 9-2+1-1=7; 3rd slot can be anything from #1 to #9 with the exception of #7 and the digit in the 2nd slot, so that's 9-1+1-2=7; so the total number of digit combinations is 7*7 = 49 numbers

If the first digit is 8: 2nd slot can be anything from #0 to #9 with the exception of 9, so that's 9-0+1-1=9; 3rd slot can be anything from #0 to #9 with the exception of #8 and the digit in the 2nd slot, so that's 9-0+1-2=8; so the total number of digit combinations is 9*8 = 72 numbers

If the first digit is 9: the combination is the same as if the first digit is 8. So, 72.

Together there are 49 + 2*72 = 193 combinations ...

What did I do wrong? Thank you!!
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Working towards 25 Kudos for the Gmatclub Exams - help meee I'm poooor

I think this is a high-quality question and I don't agree with the explanation. Question asks for three-digit integers such that all their digits are different, then below numbers should not be included right? example - 777, 788, 799, 888, 899, 877 etc..

I think this is a high-quality question and I don't agree with the explanation. Question asks for three-digit integers such that all their digits are different, then below numbers should not be included right? example - 777, 788, 799, 888, 899, 877 etc..

Yes, those numbers should not and are not included.
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Hi guys, I understand the solution above but have trouble seeing what I did wrong - can you please help?

If the first digit is 7: 2nd slot can be anything from #2 to #9 with the exception of 7, so that's 9-2+1-1=7; 3rd slot can be anything from #1 to #9 with the exception of #7 and the digit in the 2nd slot, so that's 9-1+1-2=7; so the total number of digit combinations is 7*7 = 49 numbers

If the first digit is 8: 2nd slot can be anything from #0 to #9 with the exception of 9, so that's 9-0+1-1=9; 3rd slot can be anything from #0 to #9 with the exception of #8 and the digit in the 2nd slot, so that's 9-0+1-2=8; so the total number of digit combinations is 9*8 = 72 numbers

If the first digit is 9: the combination is the same as if the first digit is 8. So, 72.

Together there are 49 + 2*72 = 193 combinations ...

Hi guys, I understand the solution above but have trouble seeing what I did wrong - can you please help?

If the first digit is 7: 2nd slot can be anything from #2 to #9 with the exception of 7, so that's 9-2+1-1=7; 3rd slot can be anything from #1 to #9 with the exception of #7 and the digit in the 2nd slot, so that's 9-1+1-2=7; so the total number of digit combinations is 7*7 = 49 numbers

If the first digit is 8: 2nd slot can be anything from #0 to #9 with the exception of 9, so that's 9-0+1-1=9; 3rd slot can be anything from #0 to #9 with the exception of #8 and the digit in the 2nd slot, so that's 9-0+1-2=8; so the total number of digit combinations is 9*8 = 72 numbers

If the first digit is 9: the combination is the same as if the first digit is 8. So, 72.

Together there are 49 + 2*72 = 193 combinations ...

What did I do wrong? Thank you!!

I also approached the question in the above way.

Bunuel, can you please explain?

Try to explain :

- You did the correct calculation for 8 and 9 hundred digit. - Anyway, you miscalculated for the 7 hundred digit : don't forget the 0 number in the ten and unit digit. That's why, the number of digits possible is 1*8*7, not 1*7*7. After this, you must add manually the number from 710-719 (which is 7 different number).