Sure, let's solve the problem step-by-step.
Given:
- [tex]\( n = 7 \)[/tex]
- [tex]\( k = 4 \)[/tex]
### 1. Calculating Permutations [tex]\( P_k^n \)[/tex]
Permutations [tex]\( P_k^n \)[/tex] represent the number of ways to arrange [tex]\( k \)[/tex] items out of [tex]\( n \)[/tex] items in a specific order. The formula for permutations is given by:
[tex]\[ P_k^n = \frac{n!}{(n-k)!} \][/tex]
For our case:
[tex]\[ P_4^7 = \frac{7!}{(7-4)!} = \frac{7!}{3!} \][/tex]
Upon evaluating this expression, we get:
[tex]\[ P_4^7 = 840 \][/tex]
So, [tex]\( P_k^n = 840 \)[/tex]
### 2. Calculating Combinations [tex]\( C_k^n \)[/tex]
Combinations [tex]\( C_k^n \)[/tex] represent the number of ways to choose [tex]\( k \)[/tex] items out of [tex]\( n \)[/tex] items without considering the order. The formula for combinations is given by:
[tex]\[ C_k^n = \frac{n!}{k!(n-k)!} \][/tex]
For our case:
[tex]\[ C_4^7 = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} \][/tex]
Upon evaluating this expression, we get:
[tex]\[ C_4^7 = 35 \][/tex]
So, [tex]\( C_k^n = 35 \)[/tex]
### 3. Calculating the Number of Ways to Select a Group of 4 out of 7 People for a Committee with 4 Distinct Roles
When selecting a group of 4 out of 7 people where there are 4 distinct roles, each selection must consider the order because the roles are distinct. Therefore, this is a permutation problem.
From our permutation calculation:
[tex]\[ \text{Number of ways} = P_4^7 = 840 \][/tex]
So, there are 840 different ways to select and assign 4 people out of 7 to distinct roles in a committee.
### Final Answers
1. [tex]\( P_k^n = 840 \)[/tex]
2. [tex]\( C_k^n = 35 \)[/tex]
The number of different ways to select a group of 4 out of 7 people for a committee in which there are 4 distinct roles is 840.
