To evaluate [tex]\( P_4^9 \)[/tex], which represents the permutation of 9 objects taken 4 at a time, we use the permutation formula:
[tex]\[ P(n, k) = \frac{n!}{(n - k)!} \][/tex]
Here, [tex]\( n \)[/tex] is 9 and [tex]\( k \)[/tex] is 4. So we need to find [tex]\( \frac{9!}{(9 - 4)!} \)[/tex].
1. Calculate [tex]\( 9! \)[/tex] (the factorial of 9):
[tex]\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880 \][/tex]
2. Calculate [tex]\( (9 - 4)! \)[/tex] (the factorial of 5):
[tex]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \][/tex]
3. Divide [tex]\( 9! \)[/tex] by [tex]\( 5! \)[/tex]:
[tex]\[ P_4^9 = \frac{9!}{5!} = \frac{362,880}{120} = 3,024 \][/tex]
So, the value of [tex]\( P_4^9 \)[/tex] is [tex]\( 3,024 \)[/tex].
To summarize:
[tex]\[ P_4^9 = 3,024 \][/tex]
