Question

Using the letters in the word TRAMPOLINE, find the number of permutations that can be formed
using 4 letters at a time. Explain how you got your answer.

Answer 1

To find the number of permutations that can be formed using 4 letters at a time from the word "TRAMPOLINE", we first need to identify the unique letters in the word.

The word "TRAMPOLINE" consists of the following letters: T, R, A, M, P, O, L, I, N, E. There are a total of 10 unique letters.

When forming permutations of 4 letters from these 10, we can use the formula for permutations, which is given by:

\[
P(n, r) = \frac{n!}{(n - r)!}
\]

Where \( n \) is the total number of unique items to choose from, and \( r \) is the number of items to choose.

In our case, \( n = 10 \) (the total number of unique letters), and \( r = 4 \) (the number of letters we want to arrange). Therefore, we can calculate the number of permutations as follows:

\[
P(10, 4) = \frac{10!}{(10 - 4)!} = \frac{10!}{6!}
\]

Now, simplifying \( 10! \) in terms of \( 6! \):

\[
P(10, 4) = \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!}
\]

The \( 6! \) cancels out:

\[
P(10, 4) = 10 \times 9 \times 8 \times 7
\]

Calculating this step-by-step:

1. \( 10 \times 9 = 90 \)
2. \( 90 \times 8 = 720 \)
3. \( 720 \times 7 = 5040 \)

Thus, the total number of permutations that can be formed using 4 letters at a time from the word "TRAMPOLINE" is **5040**.