The problem requires determining the number of 4-letter codes that can be formed using the first 10 letters of the English alphabet, considering that no letter can be repeated.

Approach:
To find the number of arrangements, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order. The formula for finding the number of permutations of n objects taken r at a time is given by:
P(n, r) = n! / (n - r)!

Given:
- First 10 letters of the English alphabet: A, B, C, D, E, F, G, H, I, J
- We need to form 4-letter codes

Calculations:
Using the formula for permutations, we can calculate the number of arrangements:

P(10, 4) = 10! / (10 - 4)!
= 10! / 6!
= (10 * 9 * 8 * 7 * 6!) / 6!
= 10 * 9 * 8 * 7
= 5,040

Therefore, there are 5,040 different 4-letter codes that can be formed using the first 10 letters of the English alphabet.

Summary:
- The number of 4-letter codes that can be formed using the first 10 letters of the English alphabet is 5,040.
- This is calculated using the formula for permutations, which takes into account the number of objects and the number of objects taken at a time.
- The formula used is P(n, r) = n! / (n - r)!, where n is the total number of objects and r is the number of objects taken at a time.