Question

use the formula for \\( _np_r \\) to evaluate the following expression.\\( _5p_3 \\)\\( _5p_3 = \square \\)

Explanation:

Step1: Recall Permutation Formula

The formula for permutations \( _nP_r \) is \( \frac{n!}{(n - r)!} \), where \( n! = n\times(n - 1)\times\cdots\times1 \) and \( 0! = 1 \). For \( _5P_3 \), \( n = 5 \) and \( r = 3 \).

Step2: Substitute Values into Formula

Substitute \( n = 5 \) and \( r = 3 \) into the formula: \( _5P_3=\frac{5!}{(5 - 3)!}=\frac{5!}{2!} \).

Step3: Expand Factorials

Expand \( 5! = 5\times4\times3\times2! \) and \( 2! = 2\times1 \). Then \( \frac{5!}{2!}=\frac{5\times4\times3\times2!}{2!} \).

Step4: Simplify Expression

Cancel out \( 2! \) from numerator and denominator: \( 5\times4\times3 = 60 \).

Answer:

\( 60 \)