can someone show me step by step??
The manager of the soda shop decides that because you are a good customer, you may use a flavor as many times as you’d like. How many different possible mixtures can you make now (remember, you have 13 flavors available to you and you must choose 6)? leave your answer in exponential or factorial form. show your work
II.) You realize that, despite what you expected, the soda actually tastes different depending on which flavor you choose first, second, etc. With this in mind, how many different sodas can you make if repetition is not allowed? Show your work and leave your answer in exponential or factorial form.
The first one is still 13C6
Review the definition of combinations to see the factorial form
Now you have 13P6
Note that 13C6 = 13P6 / 6!
is the answer 10,296?
Sure! I can help you with that.
I.) When you are able to use a flavor as many times as you'd like and you have 13 flavors to choose from, and you must choose 6 flavors, we can use the concept of combinations to determine the number of possible mixtures.
The formula for calculating the number of combinations is given by:
nCr = n! / (r!(n-r)!)
Where:
n = total number of flavors (13)
r = number of flavors to choose (6)
! = factorial
So, using this formula, we can calculate the number of different possible mixtures:
nCr = 13! / (6!(13-6)!)
= 13! / (6! * 7!)
= (13 * 12 * 11 * 10 * 9 * 8 * 7!) / (6! * 7!)
= 13 * 12 * 11 * 10 * 9 * 8
= 95,040
Therefore, you can make 95,040 different possible mixtures when repetition is allowed.
II.) However, if you now consider that the soda tastes different depending on the order of the flavors and repetition is not allowed, we can use the concept of permutations to determine the number of different sodas you can make.
The formula for calculating the number of permutations is given by:
nP_r = n! / (n-r)!
Where:
n = total number of flavors (13)
r = number of flavors to choose (6)
! = factorial
Using this formula, we can calculate the number of different sodas:
nP_r = 13! / (13-6)!
= 13! / 7!
= 13 * 12 * 11 * 10 * 9 * 8
= 95,040
Therefore, you can make 95,040 different sodas when repetition is not allowed.
So, the answer for both parts (when repetition is allowed and when repetition is not allowed) is 95,040, and the work is shown above.
To find the number of different possible mixtures when you can use a flavor as many times as you'd like (with repetition allowed), you can use the concept of combinations.
For the first question, you have 13 available flavors and you need to choose 6 flavors. In this case, repetition is allowed, meaning you can choose the same flavor multiple times.
To calculate this, you can use the formula for combinations with repetition allowed:
nCr = (n + r - 1)C(r)
Where n is the number of available options (13 flavors), and r is the number of choices you need to make (6 flavors).
So in this case, the number of combinations can be calculated as:
(13 + 6 - 1)C(6) = (18 - 1)C(6) = 17C6
Using the formula for combinations:
nCr = n! / (r! * (n - r)!)
Where "!" denotes factorial and n! represents the factorial of the number n.
So, 17C6 can be calculated as:
17! / (6! * (17 - 6)!)
Now, let's calculate this expression step by step:
1. Calculate the factorials:
17! = 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
(17 - 6)! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
2. Simplify the expression:
17! / (6! * 11!)
3. Calculate the values:
17! = 355,687,428,096,000
6! = 720
11! = 39,916,800
4. Substitute the calculated values into the expression:
355,687,428,096,000 / (720 * 39,916,800)
5. Calculate the division:
355,687,428,096,000 / 28,510,912,000 ≈ 12,487
So, the number of different possible mixtures when repetition is allowed is approximately 12,487 (in exponential form: 17C6).
Now let's move on to the second question.
If repetition is not allowed, the concept of permutations can be used to find the number of different sodas that can be made.
Again, you have 13 available flavors, but this time you need to choose 6 flavors without repetition.
The formula for permutations is:
nPr = n! / (n - r)!
So, in this case, the number of permutations can be calculated as:
13P6 = 13! / (13 - 6)!
1. Calculate the factorials:
13! = 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
(13 - 6)! = 7 * 6 * 5 * 4 * 3 * 2 * 1
2. Simplify the expression:
13! / 7!
3. Calculate the values:
13! = 6,227,020,800
7! = 5,040
4. Substitute the calculated values into the expression:
6,227,020,800 / 5,040
5. Calculate the division:
1,234,456
So, the number of different sodas that can be made when repetition is not allowed is 1,234,456 (in exponential form: 13P6).