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Bunuel
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A certain identification code consists of 6 digits, each of which is 09 Dec 2024, 00:06
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braddouglas10
O great Bunuel, the tall super attractive man, do you have any similar problems? This one seems like a complicated one for the masses!
Bunuel
A certain identification code consists of 6 digits, each of which is chosen from 1, 2, ..., 9. The same digit can appear more than once. If the sum of the 1st and 3rd digits is the 5th digit, and the sum of the 2nd and 4th digits is the 6th digit, how many identification codes are possible?

A. 625
B. 720
C. 1,296
D. 2,025
E. 6,561­
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Check Constructing Numbers, Codes and Passwords Questions in our Special Questions Directory.

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A certain identification code consists of 6 digits, each of which is 24 Aug 2025, 05:41
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And why not multiply this by 2. So 36 pairs are possible for x1 & x3 in the 6 digit code. But the 36 pairs can also be flipped => (x1,x3) = 1,8 will be different from 8,1
Here order maters right or am I missing something?

KarishmaB



I wouldn't focus on numerical answer options and to be honest, I wouldn't be able to identify that 9^4 = 6561. It is not a number I come across in GMAT much.
As for solving the question, I see that sum of 1st and 3rd digits is the 5th digit which means that sum of 1st and 3rd digits much be 9 or less.
If the 1st digit is 1, the 3rd digit could be anything from 1 to 8.
If the 1st digit is 2, the 3rd digit could be anything from 1 to 7.
and so on... We see the pattern here.
The first digit could be anything from 1 to 8 and the second digit would take 8 values or 7 values or 6 values ... 1 value.
Hence, we get 8 + 7 + ... + 1 = 8*9/2 = 36 possible values for 1st and 3rd digits. The 5th digit is defined once we pick values for 1st and 3rd digits.

The same will be applicable to 2nd, 4th and 6th digits too.

So number of ways = 36 * 36 which ends with a 6 and hence answer is (C)
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