To find the number of permutations that can be formed using 4 letters at a time from the word "TRAMPOLINE", we first need to consider the distinct letters in the word.
The word "TRAMPOLINE" consists of the following letters: T, R, A, M, P, O, L, I, N, E. There are 10 distinct letters in total.
Since we are forming permutations using 4 letters at a time from these 10 distinct letters, we can use the permutation formula:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
where \( n \) is the total number of items to choose from (in this case, 10 letters), and \( r \) is the number of items to choose (in this case, 4 letters).
Substituting \( n = 10 \) and \( r = 4 \):
\[ P(10, 4) = \frac{10!}{(10 - 4)!} = \frac{10!}{6!} \]
Now, calculating the factorial values:
\[ 10! = 10 \times 9 \times 8 \times 7 \times 6! \]
Thus, we can simplify the expression:
\[ P(10, 4) = \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!} = 10 \times 9 \times 8 \times 7 \]
Now, we calculate \( 10 \times 9 \times 8 \times 7 \):
- First, calculate \( 10 \times 9 = 90 \)
- Then \( 90 \times 8 = 720 \)
- Finally \( 720 \times 7 = 5040 \)
Therefore, the total number of permutations that can be formed using 4 letters at a time from the word "TRAMPOLINE" is:
\[ \boxed{5040} \]