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? ?

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Let $$ n = 8 $$ and $$ k = 3 $$.

1. **Permutations $$ P_{k}^{n} $$:**

The number of ways to arrange $$ k $$ out of $$ n $$ distinct items is given by:
   $$
   P(n,k) = \frac{n!}{(n-k)!}
   $$
   
   Substituting the given values:
   $$
   P(8,3) = \frac{8!}{(8-3)!} = \frac{8!}{5!}
   $$
   
   Since:
   $$
   \frac{8!}{5!} = 8 \times 7 \times 6 = 336
   $$
   
   Therefore, 
   $$
   P_{3}^{8} = 336.
   $$

2. **Combinations $$ C_{k}^{n} $$:**

The number of ways to select $$ k $$ items from $$ n $$ items (ignoring order) is given by:
   $$
   C(n,k) = \frac{n!}{k!(n-k)!}
   $$
   
   Substituting the given values:
   $$
   C(8,3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}
   $$
   
   Evaluating the expression:
   $$
   C(8,3) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56
   $$
   
   Therefore,
   $$
   C_{3}^{8} = 56.
   $$

3. **Selection of a Group for a Survey:**

The problem of selecting a group of 3 out of 8 people for a survey disregards the order of selection. Hence, the number of different ways is given by the combination:
   $$
   56.
   $$

There are 56 different ways to select a group of 3 out of 8 people for a survey.

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? ?

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