Ambiguous: Manhattan Advanced Quant - Set 10 / Prob#95

[shailendra.sharma]

Employees of a certain company are each to receive a unique 7-digit identification code consisting of the digits 0, 1, 2, 3, 4, 5, and 6 such that no digit is used more than once in any given code. In valid codes, the second digit in the code is exactly twice the first digit. How many valid codes are there?

I see a few ambiguity on this problem, and hence the solution too.

1) 7-digit code is not same as 7-digit number - that is, a number can not have highest digit as 0, but a code can have. eg: 0654321 is a valid 7 digit code, but not a a valid 7 digit number. So why the solution does not consider 0 on the highest digit.

2) What is first digit ? Is it unit digit or the highest digit ? I solved the question by assuming first digit is unit digit - I considered 0 a valid possibility, and 2nd digit as any of 1-to-6. Obviously with this I could not find any answer choice with 960 as an answer. I re-did and could get what author intended.

But in summary, I am bit confused on such ambiguities in the problem. Please clarify.

[jnelson0612]

Employees of a certain company are each to receive a unique 7-digit identification code consisting of the digits 0, 1, 2, 3, 4, 5, and 6 such that no digit is used more than once in any given code. In valid codes, the second digit in the code is exactly twice the first digit. How many valid codes are there?

I see a few ambiguity on this problem, and hence the solution too.

1) 7-digit code is not same as 7-digit number - that is, a number can not have highest digit as 0, but a code can have. eg: 0654321 is a valid 7 digit code, but not a a valid 7 digit number. So why the solution does not consider 0 on the highest digit.

0 cannot be the first digit (or the highest digit, as you say) because of the the stipulation that "In valid codes, the second digit in the code is exactly twice the first digit". If 0 were in the first position then 0 would have to also be in the second position, because 0 * 2 = 0. I can only use each digit 0 through 6 one time, according to the problem, so I can't put 0 in the first position.

2) What is first digit ? Is it unit digit or the highest digit ? I solved the question by assuming first digit is unit digit - I considered 0 a valid possibility, and 2nd digit as any of 1-to-6. Obviously with this I could not find any answer choice with 960 as an answer. I re-did and could get what author intended.

In a number, the first digit is the first number that you see when reading from left to right. For instance, in the number 543 the number "5" is the first digit. In 1234560 the number "1" is the first digit.

But in summary, I am bit confused on such ambiguities in the problem. Please clarify.

Hi Shailena,
Please see my answers above in blue.

[shailendra.sharma]

Thanks for the clarifications. It helps.

So in summary, first digit is the first digit when you read from left-to-right or the highest order digit.

[RonPurewal]

Thanks for the clarifications. It helps.

So in summary, first digit is the first digit when you read from left-to-right or the highest order digit.

well, remember -- as you said yourself! -- these are codes, not numbers.
so, there's actually no such thing as the "highest order digit", because, well, it's not a number.
there's only characters from left to right. so, the "first digit" is the same thing as, e.g., the "first letter" of a password -- which would most definitely be the letter on the left.

[shailendra.sharma]

Yep. Thanks Ron !

Need to be precise in what is code and what is number.

Code: first digit is left to right.
Number: we talk about unit digit, tens digit and so on.

Got it.

[jlucero]

Glad you got it.