Question: A 5-digit code consists of one number digit chosen from 1, 2, 3 and four letters chosen from A, B, C, D, E. If the first and last digit must be a letter digit and each digit can appear more than once in a code, how many different codes are possible?

1. 375
2. 625
3. 1,875
4. 3,750
5. 5,625

Correct Answer: E
Solution and Explanation:
Approach Solution 1:

The problem statement states that:
Given:

• A 5-digit code consists of one number digit chosen from 1, 2, 3 and four letters chosen from A, B, C, D, E.
• The first and last digit must be a letter digit and each digit can appear more than once in a code.

Find out:

• The possible number of different codes.

Please note that each digit can appear more than once in a code.
There should be 4 letters in a code (X-X-X-X) and each letter can take 5 values (A, B, C, D, E) Therefore, the total number of combinations of the letters only is 5*5*5*5=5^4.
The question states that the first and last digits must be a letter digit.
Therefore, the number digit can take any of the three slots between the letters: X-X-X-X, so 3 positions and the digit itself can take 3 values (1, 2, 3).
Hence, the total number of codes is 5^4 * 3 * 3 = 5,625.
Therefore, the possible number of different codes = 5,625.

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

• A 5-digit code consists of one number digit chosen from 1, 2, 3 and four letters chosen from A, B, C, D, E.
• The first and last digit must be a letter digit and each digit can appear more than once in a code.

Find out:

• The possible number of different codes.

Select one number from 3 numbers in 3C1 = 3 ways
Select one position from the middle three for the number in 3C1 = 3 ways
The other four positions can be filled by the 5 letters in 5^4 ways.
Therefore, the total number of codes possible = 3 * 3 * (5^4) = 5,625

Approach Solution 3:
The problem statement implies that:
Given:

• A 5-digit code consists of one number digit chosen from 1, 2, 3 and four letters chosen from A, B, C, D, E.
• The first and last digit must be a letter digit and each digit can appear more than once in a code.

Find out:

• The possible number of different codes.

Since repetitions are allowed, there are 5 ways to select the letters for the first slot and five ways to pick for the last slot
Therefore, the sequence is 5 x _ x _ x _ x 5
We still need to choose 1 digit and 2 letters between.
For the letters, again, repetitions are allowed so each slot can be filled in 5 ways.
For digit, it can only be filled in 3 ways.\
However, since the digit can be in any one of the 3 middle slots, so we need to multiply by 3 again:
5 x 3 x 5 x 5 x 5 or
5 x 5 x 3 x 5 x 5 or
5 x 5 x 5 x 3 x 5
Therefore, the total number of codes possible = 5 x 5 x 5 x 5 x 3 x 3 = 5^4 x 3^2 = 5625.
Hence, the total number of codes possible = 5625

“A 5-digit code consists of one number digit chosen from 1, 2, 3”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Advanced Questions”. The candidate needs to analyse every data of the GMAT Problem Solving questions in order to solve quantitative problems. GMAT Quant practice papers help the candidates to get familiar with different types of questions that will improve their mathematical knowledge.

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