Laura wants 3 scoops of ice cream. She can choose from vanilla, chocolate, mint, raspberry, blueberry, and coffee flavors.
How many combinations of 3 scoops of ice cream can Laura choose from? and is this a scenario in which repetition is allowed?
Case 1: she picks 3 different flavours, does not care in which order they are picked,
= C(6,3) = 20
(you could actually list those)
Case 2: repetition is allowed, order does not matter, e.g. {mint,mint,vanilla} or {ras,ras,ras}
triples: choose one of the 6 colours, C(6,1) = 6
double and single: choose the double flavour ---> 6
leaving 5 to choose for the single flavour
total of that case =30
total = 20 + 30 + 6 = 56
Now if we also cared in which order the flavours are put in the cone, it would change the argument.
Well, Laura is about to embark on a serious ice cream adventure! Let's figure out how many flavor combinations she can create.
Since Laura can choose from 6 different flavors, she has 6 options for her first scoop. After that, she still has 6 options for her second scoop, and again 6 options for her third scoop.
To find the total number of combinations, we multiply the number of options for each scoop together:
6 * 6 * 6 = 216
Therefore, Laura can choose from 216 different combinations of 3 scoops of ice cream.
Now, as for repetition, it seems like Laura is allowed to have the same flavor multiple times in her combination. So, if she wants to have all three scoops of vanilla, she can do that!
Hope Laura enjoys her ice cream extravaganza!
To find the number of combinations of three scoops of ice cream that Laura can choose from, we need to consider whether repetition is allowed.
1. If repetition is allowed:
In this case, Laura can choose the same flavor for more than one scoop. Since she has 6 flavors to choose from and repetition is allowed, she can choose any flavor for each of the three scoops.
The number of possible combinations can be calculated using the formula:
Number of combinations = (number of options)^(number of choices)
In this case, since Laura has 6 options (flavors) and she needs to choose 3 scoops, we can calculate the number of combinations as follows:
Number of combinations = 6^3 = 216
Therefore, there are 216 possible combinations of 3 scoops of ice cream that Laura can choose from when repetition is allowed.
2. If repetition is not allowed:
In this case, Laura can choose each flavor only once for the three scoops.
The number of possible combinations can be calculated using the formula for combinations:
Number of combinations = n! / (r! * (n-r)!)
Where n is the total number of options (flavors) and r is the number of choices (scoops).
In this case, n = 6 (flavors) and r = 3 (scoops).
Substituting the values:
Number of combinations = 6! / (3! * (6-3)!)
= 6! / (3! * 3!)
= (6 * 5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (3 * 2 * 1))
Simplifying the expression:
Number of combinations = (6 * 5 * 4) / (3 * 2 * 1)
= 20
Therefore, there are 20 possible combinations of 3 scoops of ice cream that Laura can choose from when repetition is not allowed.
In summary, if repetition is allowed, there are 216 possible combinations, and if repetition is not allowed, there are 20 possible combinations of 3 scoops of ice cream that Laura can choose from.
To find how many combinations of 3 scoops of ice cream Laura can choose from, we can use the concept of combinations.
Combinations are used when the order of the elements does not matter. In this scenario, the order in which Laura chooses the scoops of ice cream does not matter.
Since repetition is allowed, Laura can choose the same flavor multiple times.
To calculate the number of combinations, we can use the formula for combinations with repetition allowed:
n + r - 1 C r
Where:
n is the number of options available to choose from
r is the number of choices to be made
In this case, n = 6 (vanilla, chocolate, mint, raspberry, blueberry, coffee) and r = 3 (number of scoops Laura wants).
Plugging in the values into the formula, we get:
6 + 3 - 1 C 3 = 8 C 3
To calculate the value of 8 C 3, we need to use the formula for calculating combinations:
n! / (r! * (n - r)!)
Where "!" denotes factorial.
8! / (3! * (8 - 3)!)
Calculating the factorials:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320
3! = 3 * 2 * 1 = 6
(8 - 3)! = 5! = 5 * 4 * 3 * 2 * 1 = 120
Plugging in the values:
40320 / (6 * 120) = 40320 / 720
Simplifying the fraction:
= 56
Therefore, Laura can choose from 56 different combinations of 3 scoops of ice cream when repetition is allowed.