The difference between combinations and permutations

When to Use Which:

• Use Permutations when the order of selection is important. This often involves situations with rankings, specific roles, or arrangements.
• Use Combinations when the order of selection is NOT important. This usually involves selecting a group or subset where the arrangement within the group doesn't matter.  

Now, let's analyze each selection:

a) Combination: You are selecting 3 truck models out of 8 to test drive. The order in which you test drive them doesn't change the group of trucks you're considering.

b) Permutation: Electing a president and vice-president from 12 members involves order because the roles are distinct. Sarah as president and John as vice-president is a different outcome than John as president and Sarah as vice-president.

c) Combination: Choosing 9 fans out of 9 to meet the team doesn't involve any specific order or ranking among the chosen fans.

d) Combination: Selecting 2 class representatives from 120 seniors is a combination because the order in which they are chosen doesn't define different roles; they are simply two representatives.

e) Permutation: Choosing a pitcher and 4 infielders involves order because the pitcher has a specific role distinct from the infielders. The selection of a particular player as pitcher versus as an infielder creates a different team composition.

3. Let's calculate the probabilities:

   The total number of ways to select a committee of three from the 10 city council members is given by the combination formula: C(n,k)=k!(n−k)!n!​ Where $n$ is the total number of items, and $k$ is the number of items to choose.  

   So, the total number of possible committees is: C(10,3)=3!(10−3)!10!​=3!7!10!​=3×2×110×9×8​=10×3×4=120 There are 120 possible committees of three.

   All Republicans: There are 4 Republicans. The number of ways to select a committee of three Republicans from the four is: C(4,3)=3!(4−3)!4!​=3!1!4!​=(3×2×1)(1)4×3×2×1​=4 The probability of selecting a committee of all Republicans is the number of ways to select all Republicans divided by the total number of possible committees: P(All Republicans)=Total number of possible committeesNumber of ways to select 3 Republicans​=1204​ Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor (4): P(All Republicans)=120÷44÷4​=301​

   All Democrats: There are 3 Democrats. The number of ways to select a committee of three Democrats from the three is: C(3,3)=3!(3−3)!3!​=3!0!3!​=(3×2×1)(1)3×2×1​=1 (Remember that $0!=1$) The probability of selecting a committee of all Democrats is the number of ways to select all Democrats divided by the total number of possible committees: P(All Democrats)=Total number of possible committeesNumber of ways to select 3 Democrats​=1201​

   The probability of selecting all Republicans is 301​​. The probability of selecting all Democrats is 1201​​.