Certainly! To determine how many different four-letter permutations can be formed using letters out of the first twelve letters in the alphabet, we will use the permutation formula given by:
[tex]\[ _nP_r = \frac{n!}{(n-r)!} \][/tex]
Here, [tex]\( n \)[/tex] represents the total number of items to choose from, and [tex]\( r \)[/tex] represents the number of items to arrange.
Step-by-Step Solution:
1. Identify the values for [tex]\( n \)[/tex] and [tex]\( r \)[/tex]:
- [tex]\( n = 12 \)[/tex] (since we are choosing from the first twelve letters of the alphabet)
- [tex]\( r = 4 \)[/tex] (since we want to form four-letter permutations)
2. Substitute the values into the permutation formula:
[tex]\[ _nP_r = \frac{12!}{(12-4)!} = \frac{12!}{8!} \][/tex]
3. Calculate the factorials:
[tex]\[ 12! = 12 \times 11 \times 10 \times 9 \times 8! \][/tex]
Notice that [tex]\( 8! \)[/tex] in the numerator and denominator cancel out, simplifying the expression to:
[tex]\[ \frac{12 \times 11 \times 10 \times 9 \times 8!}{8!} = 12 \times 11 \times 10 \times 9 \][/tex]
4. Compute the multiplication:
[tex]\[ 12 \times 11 = 132 \][/tex]
[tex]\[ 132 \times 10 = 1320 \][/tex]
[tex]\[ 1320 \times 9 = 11880 \][/tex]
So, the number of different four-letter permutations that can be formed using any four letters out of the first twelve letters in the alphabet is:
[tex]\[ \boxed{11880} \][/tex]
