Question: A four-character password consists of one letter of the alphabet and three different digits between 0 and 9, inclusive. The letter must appear as the second or third character of the password. How many different passwords are possible?

(A) 5,040
(B) 18,720
(C) 26,000
(D) 37,440
(E) 52,000

Correct Answer: D
Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• A four-character password consists of one letter of the alphabet and three different digits between 0 and 9, inclusive.
• The letter must appear as the second or third character of the password.

Find out:

• The number of different passwords is possible.

Let the 4-character password be DLDD.
The first digit can be chosen in 10 ways.
The letter can be chosen in 26 ways.
The next digit can be chosen in 9 ways.
The next digit can be chosen in 8 ways.
Therefore, this gives us 10 * 26 * 9 * 8 ways
Now, the letter can also be DDLD so there will be another 10 * 9 * 26 * 8 ways
Total = 10 * 26 * 9 * 8 * 2 = 37,440 ways
Therefore, the number of ways the different passwords are possible = 37,440 ways

Approach Solution 2:
The problem statement informs that:
Given:

• A four-character password consists of one letter of the alphabet and three different digits between 0 and 9, inclusive.
• The letter must appear as the second or third character of the password.

Find out:

• The number of different passwords is possible.

Let the Password be: _ _ _ _
We know that Letters are a Total of 26.
We need to use 1 at the position: 2nd or 3^rd
Digits: Total 10 [from 0 to 9]: Different digits need to be used.
Case I: The letter in the 2nd position will have 26 options.
Digits: [1st place: 10 options ] * [3rd place: 9 options ] * [4th place: 8 options ]
Therefore, the 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 altered not the options available]
The letter in 3rd position will have 26 options.
Digits: [1st place: 10 options ] * [2nd place: 9 options ] * [4th place: 8 options ]
Total ways: 10 * 9 * 26 * 8 = 18,720.
Hence, the overall ways are 18,720 + 18,720 = 37,440.
Therefore, the number of ways the different passwords are possible = 37,440 ways

Approach Solution 3:
The problem statement implies that:

Given:

• A four-character password consists of one letter of the alphabet and three different digits between 0 and 9, inclusive.
• The letter must appear as the second or third character of the password.

Find out:

• The number of different passwords is possible.

We need to use 1 Letter - out of 26 Letters in the alphabet. We must go into either slot 2 or slot 3
Then we have to choose 3 Digits out of 10 Options to fill the remaining slots. We should arrange each combination of 3 Digits chosen
Scenario 1: Letter Appears in Slot 2
“26 choose 1” for Slot 2 i.e. 26 Options that can go in Slot 2.
Only 1 way to organise the Letter selected because the 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 i.e we can say 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
Therefore, from Scenario 1, we get:
Ways = (26) * (720) = 18, 720
Scenario 2: Letter Appears in Slot 3
“26 choose 1” for Slot 3 i.e. 26 Options that can go in Slot 3.
Only 1 way to organise the Letter selected because the Letter is anchored in Slot 3.
It must stay in Slot 3.
26 * 1 = 26
And “10 choose 3” for the other 3 remaining Slots i.e we can say 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
Therefore, from Scenario 2, we get:
Ways = (26) * (720) = 18, 720
That is we get another 18, 720 ways
The number of ways the different passwords are possible = (2) * (18, 720) = 37, 440.

“A four-character password consists of one letter of the alphabet”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “Cracking the GMAT with 2 Practice Tests, 2014 Edition”. The GMAT Problem Solving questions required good skills in calculations in order to solve quantitative problems. GMAT Quant practice papers enable the candidates to practice lots of questions that will help them to improve their quantitative knowledge.

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