- How many three-digit numbers are so that they have 2 equal

(1) assuming the first digit cannot be zero, 243
(2) 42
(3) both A and B as stated in your problem. (I beileve b is a typo and should be 165 -- then the answer is A).

Akamaibrah,

I disagree with your answers on (2) and (3)

(2) I think it should be 21
(3) I think the right answer is 163 - choice (b) since the rest are all divisible by 11. The working is as follows:

Lets assume M has the digits x and y. Therefore, M=10x+y
This gives us N=10y+x
Sum of the two gives M+N=11(x+y) and the answer follows from this.

Comments?

Yes I realize that the number must be divisible by 11. However, NEITHER 181 nor 163 is divisible by 11 (here is a quick check. Subtract the sum of the digits in the odd-positions from the sum of the digits in the even positions. If the answer is 0 or a multiple of 11, then the whole number is divisible by 11). 165 would be. Or 187. So there must be a typo in one of those two. Hence, my answer.

817
826
835
844
853
862
871
718
727
736
745
754
763
772
781
628
637
646
655
664
673
682

well, i'm up to 22 and there seems to be a few numbers left....
I'll stand by 42 for now.



P.S. If order doesn't matter, then the answer is 9.

Agree with you on both....

my bad, agree. the third 9 is out.

3*9*9*1=243

Question 1: What is the systamatic and faster way to slove this question?

Question 2: My ans is 42. But, how to get hte answer in 2 minutes?

Question 3: I agree that there is a typo in the answer chices. Same
question appaered in a different forum couple of days back with the
same typo?!

1) There are 10 possible number pairs that can combine with 9 other numbers = 90. Of each of these, there are 3 ways that they can be arranged: AAB ABA BAA. So we now have 90 x 3 = 270. But we need to exclude all of the numbers starting with zero. Since this is a "symmetric" problem, exactly 1/10 of all possible numbers will start with zero so the answer is 270 x 90% = 243.

2) Consider 3 number like 3 dice. Let say we look at the first die. Given that the first die is an 8, the next two must add up to 16. Using the same logic as dice, you can quickly see that there are 7 ways to combine two 8-sided dices to make 8.
If the first die is 7, then the next 2 must add up to 9 and there are 8 ways to do that.
If the first die is 6, then the next 2 must add up to 10 and there are 7 ways to do that. Similarly :
1st = 5, ways to make 11 = 6
1st = 4, ways to make 12 = 5
1st = 3, wyas to make 13 = 4
1st = 2, ways to make 14 = 3
1st = 1, wyas to make 15 = 2
If you spot the pattern and understand how two dice combine, you can quickly fill in this table knowing only the first 3 results.

Hence the total is 7 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 42

please... could you give me a systamatic way for 2?

I guess 243 is not the answer. Considering digits from 1-9, we get 3 x 9 x 8 x 1. When zero is in the last digit, 9 combinations and when zero is in the second digit, 9 more of them. Hence I get 234 as the answer.

When you write 3 x 9 x 9 x 1, you are allowing 0 to occur in all the three places which means it may not be a three digit no. at all.

Bharathi.

the answer is 243. Stolyars method is poor and he was "lucky" to stumble on the correct answer.