Wildflower wrote:
For a 3 digit number to be even, it has to be either 0 or 6.
Constraint: all digits have to be different
Case 1: With 0 as the last digit.
Units digit = 0 --> 1 way
Tens digit can be 3,1,6,7,9, and NOT 0 --> 5 ways
Hundreds digit cannot be 0 and tens digit --> 4 ways
Total = 1 x 5 x 4 = 20 ways
Case 2: With 6 as the last digit
Units digit = 6 --> 1 way
Tens digit cannot be 6 --> 5 ways
Hundreds digit cannot be units digit, tens digit, and zero --> 3 ways
Total = 1 x 5 x 3 = 15 ways
Total = case 1 + case 2 = 20 + 15 = 35 ways.
Could someone please tell me where I went wrong? Thank you!
Hey
Wildflower,
I have highlighted the areas where you made a mistake.
Whenever you have a situation in which 0 is involved, try to fill that space first, where the confusion will happen (hundreds place in this case)!
When 6 has been placed in the units digit, we are left with the following options for tens and hundreds place: 0, 1, 3, 7 and 9
Now, if you fill the tens place first and say that there are five ways to fill it, then you are basically saying that I can put 0 in the hundreds place, I can put 1 also, I can put 3 or 7 or 9 also in the hundreds place.
Now think a bit. If you put 6 in the units place and say 0 in the tens place, how many digits are available for the hundreds place? We have 3,7,9 and 1 available right?
However, you have written that the hundreds place can be filled in only 3 ways, which is incorrect!
So, I hope you understand what cases you are missing out?
You are missing those cases, when 0 is put on the tens place.
Thus, to avoid such confusion, what we do is that we fill the hundreds place first and say that the hundreds place can be filled in 4 ways (1,3,7 or 9), this ensure that all eligible digits will get a chance to be placed at the hundreds place.
Now with are left with 0 and the remaining 3 digits (since one of the digits is already placed at the hundreds place), thus total available cases for the tens place will also be 4 and thus the correct answer will be 4 x 4 = 16.
Let me know if you still have any doubts.
Regards,
Saquib
Quant Expert
e-GMATI see where I went wrong. Thank you,