Compute the following for \( n=8 \) and \( k=3 \) : i. \( P_{k}^{n}= \) ii. \( C_{k}^{n}= \) How many different ways are there to select a group of 3 out of 8 people for a survey? ?

To find \( P_{3}^{8} \), we calculate the number of permutations of selecting 3 people out of 8, which is given by the formula \( P_{k}^{n} = \frac{n!}{(n-k)!} \). Therefore, \( P_{3}^{8} = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336 \). For \( C_{3}^{8} \), we find the number of combinations of selecting 3 people out of 8, which is computed using the formula \( C_{k}^{n} = \frac{n!}{k!(n-k)!} \). Thus, \( C_{3}^{8} = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \). So, there are 336 different ways to arrange 3 people and 56 different ways to select a group of 3 out of 8 people for the survey!