A four-character password consists of one letter of the alphabet

Hi peggydia,

The prompt tells us that the code has to use "three DIFFERENT digits", so duplicates are NOT allowed.

As you continue to practice, you have to take careful note of the specific 'rules' that each prompt gives you to work with. On Test Day, it's common for one or more of the wrong answers to be based on a specific detail that a Test Taker might miss.

GMAT assassins aren't born, they're made,
Rich

GMAT questions will not ask you to guess/assume. If you are required to take distinct digits, it will be mentioned (as done in this question by writing "different digits" ). If nothing is given, then you take the most general case of allowing repetition in such questions.

Since order matters here, you should multiply the number of possibilities for each character in the password.

Case 1: Letter is second

You have DLDD. Note that the digits need to be different from one another, so the numbers go down.

> Combinations = 10 × 26 × 9 × 8

Case 2: Letter is third
This will be the same thing, just with the order of the middle numbers changed.

Answer
Add the two values. Since they are the same, you can just multiply by 2.

> Answer = 2 × 10 × 26 × 9 × 8 = 36,770

Have to use 1 Letter - out of 26 Letters in the alphabet - to go into either slot 2 or slot 3


then have to choose 3 Digits out of 10 Options to fill the remaining slots and Arrange each combination of 3 Digits chosen




Scenario 1: Letter Appears in Slot 2


"26 choose 1" for Slot 2 ---- 26 Options that can go in Slot 2.
Only 1 way to arrange the Letter chosen because Letter is anchored in Slot 2. It must stay in Slot 2.

26 * 1 = 26

AND


"10 choose 3" for the other 3 remaining Slots ---- 10! / (3! 7!)

then need to arrange those digits in those slots in 3! ways

10! / (3! 7!) * 3! = 10! / 7! = 10 * 9 * 8 * 7! / 7! = 10 * 9 * 8 = 720


Scenario 1 Ways -- (26) * (720) = 18, 720


Scenario 2: Same logic as above, except the Letter is now anchored in Slot 3.

another 18, 720 ways




(2) * (18, 720) = 37, 440

-D-

Password: _ _ _ _

Letter: Total 26. To be used: 1 at the position: \(2^{nd}\) or \(3^{rd}\)

Digits: Total 10 [from 0 to 9]: Different digits to be used.

Case I: Letter at \(2^{nd}\) position will have 26 options.

Digits: [\(1^{st}\) place: 10 options ] * [\(3^{rd}\) place: 9 options ] * [\(4^{th}\) place: 8 options ]

Total ways: 10 * 26 * 9 * 8 = 18,720.


Case II: It will also give the same total of 18,720 [Just the position of the letter has changed not the options available]

Letter at \(3^{rd}\) position will have 26 options.

Digits: [\(1^{st}\) place: 10 options ] * [\(2^{nd}\) place: 9 options ] * [\(4^{th}\) place: 8 options ]

Total ways: 10 * 9 * 26 * 8 = 18,720.

Overall ways: 18,720 + 18,720 = 37,440

Answer D

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