To determine how many ways we can arrange eight letters into groups of five where order matters and the first two letters are already chosen, let's break down the problem step-by-step:
1. Step 1: Understand the problem
- We have eight letters in total.
- We need to create groups of five letters.
- The first two letters of these groups are already chosen and fixed.
- So, we only need to determine how to arrange the remaining letters.
2. Step 2: Calculate the number of remaining letters
- Since the first two letters are already chosen, we have [tex]\(8 - 2 = 6\)[/tex] letters remaining to choose from.
3. Step 3: Determine the number of letters to arrange
- We need to arrange [tex]\(5 - 2 = 3\)[/tex] more letters out of the six remaining.
4. Step 4: Calculate the number of ways to arrange the remaining letters
- We are arranging 3 letters out of 6 in a specific order, which is equivalent to finding permutations of 6 taken 3 at a time.
- The number of permutations of [tex]\(n\)[/tex] items taken [tex]\(r\)[/tex] at a time is given by the formula: [tex]\(P(n, r) = \frac{n!}{(n-r)!}\)[/tex].
- Plugging in our values, we get [tex]\(P(6, 3)\)[/tex].
5. Result
- Using the values given in the result, we know there are 120 ways to arrange the remaining letters.
Thus, the number of ways to arrange eight letters into groups of five where the order matters and the first two letters are already chosen is [tex]\(120\)[/tex].
So, the correct answer is 120.
