Evaluate the expression. \( { }_{9} P_{5} \) A. 15,120 B. 504 C. 1 D. 9
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To evaluate the expression \( { }_{9} P_{5} \), we use the formula for permutations, which is given by: \[ _{n}P_{r} = \frac{n!}{(n-r)!} \] In this case, \( n = 9 \) and \( r = 5 \). Now substituting the values: \[ _{9}P_{5} = \frac{9!}{(9-5)!} = \frac{9!}{4!} \] Calculating the factorials, we have: \[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4! \] Thus the \( 4! \) in the numerator and denominator cancel each other out: \[ _{9}P_{5} = 9 \times 8 \times 7 \times 6 \times 5 \] Calculating this step by step: 1. \( 9 \times 8 = 72 \) 2. \( 72 \times 7 = 504 \) 3. \( 504 \times 6 = 3024 \) 4. \( 3024 \times 5 = 15120 \) Therefore, \( { }_{9} P_{5} = 15,120 \). The correct answer is **A. 15,120**.