Leah has a list of 8 cities to visit on a road trip with some friends. She needs to finalize the plans for the trip. In how many different orders can she plan to visit the cities?
orders
There are 40,320 different orders Leah can plan to visit the cities.
This can be calculated using the formula for permutations of n objects:
n! = n x (n-1) x (n-2) x ... x 2 x 1
In this case, n = 8 cities.
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320.
So there are 40,320 different orders Leah can plan to visit the cities on her road trip.
Logan is giving an oral presentation on different modes of transportation. He wants to cover buses, horses, and motorcycles. In how many different orders can he include them in his presentation?
orders
There are 6 different orders Logan can include the modes of transportation in his presentation.
This can be calculated using the formula for permutations of n objects:
n! = n x (n-1) x (n-2) x ... x 2 x 1
In this case, n = 3 modes of transportation.
3! = 3 x 2 x 1 = 6.
So there are 6 different orders Logan can include buses, horses, and motorcycles in his presentation.
Some examples of the different orders are:
1. Buses, horses, motorcycles
2. Buses, motorcycles, horses
3. Horses, buses, motorcycles
4. Horses, motorcycles, buses
5. Motorcycles, buses, horses
6. Motorcycles, horses, buses
After moving into a new apartment, Terrell decides to line up mementos on the mantle of the fireplace. If he has 8 mementos to choose from, in how many ways can he pick 2 and line them up?
ways
There are 28 ways Terrell can pick 2 mementos from 8 and line them up.
This can be calculated using the formula for combinations of n objects taken k at a time:
nCk = n! / (k! (n-k)!)
In this case, n = 8 mementos and k = 2 mementos to choose and line up.
8C2 = 8! / (2! (8-2)!)
8C2 = 8! / (2! 6!)
8C2 = (8 x 7 x 6!) / (2 x 1 x 6!)
8C2 = 28
So there are 28 ways Terrell can pick 2 mementos from 8 and line them up on the mantle.
There are 9 employees at a corporate retreat, but a particular team building exercise requires only 2 people, each playing a different role. How many ways are there to select 2 employees and assign them to the roles?
ways
There are 72 ways to select 2 employees and assign them to the roles.
This can be calculated using the formula for combinations of n objects taken k at a time:
nCk = n! / (k! (n-k)!)
In this case, n = 9 employees and k = 2 employees to choose and assign to roles.
9C2 = 9! / (2! (9-2)!)
9C2 = 9! / (2! 7!)
9C2 = (9 x 8 x 7!) / (2 x 1 x 7!)
9C2 = 36
However, we must multiply by the number of ways to assign each selected pair of employees to the two roles. There are 2 options for the first role and 1 option left for the second role. So we multiply 36 by 2 to get the total number of ways to select 2 employees and assign them to the roles:
36 x 2 = 72
Therefore, there are 72 ways to select 2 employees and assign them to the roles for the team building exercise.
A band knows how to play 5 different songs. How many different 2-song set lists could the band perform at their next concert?
set lists
There are 10 different 2-song set lists the band could perform at their next concert.
This can be calculated using the formula for combinations of n objects taken k at a time:
nCk = n! / (k! (n-k)!)
In this case, n = 5 songs and k = 2 songs to choose for the set list.
5C2 = 5! / (2! (5-2)!)
5C2 = 5! / (2! 3!)
5C2 = (5 x 4 x 3!) / (2 x 1 x 3!)
5C2 = 10
So there are 10 different 2-song set lists the band could perform at their next concert.
Another way to think about this is to use the formula for permutations of n objects taken k at a time:
nPk = n! / (n-k)!
In this case, we want to find the number of 2-song set lists, which is the same as arranging 2 songs from 5 in order.
5P2 = 5! / (5-2)!
5P2 = 5! / 3!
5P2 = (5 x 4 x 3!) / 3!
5P2 = 20 / 2
5P2 = 10
So again, there are 10 different 2-song set lists the band could perform at their next concert.
During recess, a child is playing with a bag of 5 marbles, all in different colors. She randomly takes 3 marbles out of the bag, one at a time. How many sequences of 3 marbles are possible?
sequences