Calculate Combinations of 10 Items (Max 3) - Paul

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In summary, the problem is asking how many different sandwich options can be made with a menu consisting of 3 breads, 3 meats, 4 cheeses, and 8 toppings, with the restriction of only one type of bread, up to one meat, up to two cheeses, and up to three toppings per sandwich. Using the fundamental principle of counting and combinations, the total number of combinations is 120. The problem can be solved in three cases: bread, bread and meat, and bread, meat, and cheese. Adding these three cases together gives the total number of combinations.
  • #1
paulhunn
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Say i needed to calculate the different number of combinations there are if you have 10 items and can pick up to 3 of them. e.g you buy a sandwhich and have Ketchup, Mustard, Relish, Lettuce, Pickles, Sour Cream, Cream Cheese, Olives as available toppings but you can only choose up to three. how many combinations are there?

Thanks
Paul
 
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  • #2
What have you done so far to solve your problem?
 
  • #3
Well the entire problem is as follows:
You are at a corner deli with a craving for a sandwich, here's the menu:
Breads: wheat, rye, white
Meats: turkey, ham, salami
Cheeses: American, Swiss, Cheddar, Gouda
Toppings: Ketchup, Mustard, Relish, Lettuce, Pickles, Sour Cream, Cream Cheese, Olives

You can only get one kind of bread (you have to have bread, no low-carb diet).
Per sandwich you're only allowed up to one kind of meat, up to two kinds of cheese, and up to three toppings. This means you can have none of the above options. The minimum required food is bread with nothing on it.

How many different options do you have?



So far i have worked out that there are 120 combinations up to the cheese selection (unless i have made an error)
 
  • #4
Use combination and fundamental principle of counting
 
  • #5
think of how many choices you have for the each selection. Then you can use the rule of product or the formula P(n,r)=n!/(n-r)! might help you out.
 
  • #6
buzzmath said:
think of how many choices you have for the each selection. Then you can use the rule of product or the formula P(n,r)=n!/(n-r)! might help you out.

I don't think this would be a situation to use a permutation, a combination would be better, i think the problem would consider a sandwich with lettuce and chese to be the same as a sandwich with cheese and lettuce, so order doesn't matter. Don't read too much into that example because it doesn't really fit what the problem is asking but it gets my point across..
 
  • #7
Solve in three cases. In each case you can use combination and further you can add the three cases to get the result.
 

1. What does "calculate combinations of 10 items (max 3)" mean?

This phrase refers to determining the number of different ways that 10 items can be combined in groups of 3. It is a mathematical concept that is used in various fields such as statistics, probability, and computer science.

2. How do I calculate combinations of 10 items (max 3)?

To calculate combinations of 10 items (max 3), you can use the formula nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items in each combination. In this case, n = 10 and r = 3. So the calculation would be 10C3 = 10! / 3!(10-3)! = 120 different combinations.

3. What is the difference between combinations and permutations?

Combinations and permutations are both ways of arranging items, but they differ in the order of the items. In combinations, the order does not matter, while in permutations, the order does matter. For example, the combination "red, blue, green" would be the same as "green, blue, red", but in permutations, these would be considered as two different arrangements.

4. How is the concept of combinations used in real-life situations?

The concept of combinations is used in various real-life situations, such as in probability and statistics to calculate the likelihood of events. It is also used in the field of genetics to determine the possible combinations of genes in offspring. In computer science, combinations are used in algorithms for data compression and encryption.

5. What are the limitations of calculating combinations of 10 items (max 3)?

The main limitation of calculating combinations of 10 items (max 3) is that it only gives the number of possible combinations, but does not provide any information about the specific combinations. Also, the formula becomes more complex and time-consuming to use for larger numbers. Additionally, it assumes that all items are unique and can only be used once in a combination.

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