For these two review questions, I want to make sure I'm understanding the concept behind these.
1) A dessert stand sells candy bars for $2.00 and ice cream sandwiches for $3.00. If 100 of these were sold for $250, how many candy bars and ice cream sandwiches are sold? Set up the equations and solve
I think I am setting up these two as systems of linear equations right?
2) I have five different hats and 10 different coats. I want to take 3 of the hats and 7 of the coats with me on a trip. How many ways can I choose these items? Set up only.
Is this a permutations/combinations problem or a multiplication principle problem?
1. Candy bars -- x
sandwiches -- y
x + y = 100
2x + 3y = 250
I don't know what method you have learned to solve these,
elimination: I would multiply the first equation by 2 and subtract them
or
substitution: from the first, y = 100-x, now sub that into the 2nd
2. I don't think the order in which you choose the hats and the coats matters, so this is a combination.
So for the hats ----> C(5,3)
for the coats -----> C(10,7)
multiply them
1) Yes, you are correct. To solve this problem, you can set up a system of linear equations. Let's denote the number of candy bars as 'x' and the number of ice cream sandwiches as 'y'.
Since the dessert stand sells candy bars for $2.00, the total cost of candy bars sold can be represented as 2x. Similarly, the total cost of ice cream sandwiches sold can be represented as 3y.
We are given that 100 of these items were sold for $250. Therefore, we can set up the following equation:
2x + 3y = 250
Additionally, we are also given that the total number of items sold is 100. So we can set up another equation:
x + y = 100
Now you have a system of linear equations:
2x + 3y = 250
x + y = 100
Solving this system of equations will give you the values of x and y, which represent the number of candy bars and ice cream sandwiches sold.
2) This is a combinations problem.
To determine the number of ways you can choose 3 hats out of 5 and 7 coats out of 10, you need to use combinations since the order in which you choose the items doesn't matter.
The number of ways to choose 3 hats out of 5 can be calculated using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
For this case, n (the total number of hats) is 5, and r (the number of hats you want to choose) is 3.
Similarly, the number of ways to choose 7 coats out of 10 can be calculated using the same formula, where n (the total number of coats) is 10, and r (the number of coats you want to choose) is 7.
However, to set up the problem, you don't actually need to calculate the exact number of combinations. You only need to express the problem using the combination notation:
C(5, 3) * C(10, 7)
This represents the total number of ways you can choose 3 hats out of 5 and 7 coats out of 10.