Difference between at least and exactly? combinatorics

In summary, the conversation discusses two problems involving choosing different collections of 20 coins with specific criteria. The first problem requires at least 4 pennies to be chosen and has 4 categories (quarters, dimes, nickels, pennies), resulting in 969 possible collections. The second problem requires exactly 4 pennies to be chosen and has 3 categories (quarters, dimes, nickels), resulting in 153 possible collections. This is because in the second problem, the remaining 16 coins cannot be pennies, so the number of categories is reduced to 3.
  • #1
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Hello everyone.

I'm revewing these 2 problems and im' not seeing how they are getting the answer for the "...exactly 4 penny's"

Here's the 2 problems:

(a) A large pile of coins consist of quarters, dimes, nickels, and pennies (at least 20 of each). How many different collections of 20 coins can be chosen if AT LEAST 4 pennies must be chosen?

Well you have 4 categories, Quaters, Dimes, nickles and Pennies it says (at least 20 of each).

I'm going to represent the categories as bars |'s and the coins as x's.
If the problem said, if at least 4 pennies are already chosen you can assume you already put 4 x's in the penny category, now you have 16 places left to put coins, anywhere you want.

so you could have like this:
xxxxx|xxxxx|xxxx|xx
-------------------
xxxx
P N Q D
The above ---- means the coins u distrubted, and the stuff under the --- means the coins already there.
So if you have 4 categories that's represented by n-1 bars (3) and 16 x's. So you have a total of

(16+3 choose 16) = 969 which is the correct answer.


Now for the second question:
(b) How many different collections of 20 coins can be chosen if EXACTLY 4 pennies must be chosen?

THe work shown is, they only used 2 bars |'s and 16 x's and got:

(16 + 2 choose 16) = (18 choose 16) = 153.

Now why did they only have 3 categories now isntead of 4? The reason I say they only had 3 is because they had 2 bars, thus they had to have 2+1 categories.

Why in the first problem did they have to use 4 categories, is it because they said "at least 20 of each?"

and in the 2nd problem they only said, If you have exactly 4 penny's, how many different collections of 20 coins can be chosen?

I thought to myself, well if only 4 penny's are allowed the minimum amount of categories to use and still put 20 coins in would be to have a category of Penny's which is going to have 4 x's, now your going to have 16 x's (coins) to put in any category you want. So why couldn't u put 4 x's in penny's and 16 in dimes?

Then you would only have 2 categories, so 2-1 = 1 bar. Then you would have

(16 + 1 choose 16) = (17 choose 16)

But the answer is (16+2 choose 16) = (18 choose 16), why did they have to use 3 categories?


THanks!
 
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  • #2
The reason is this. In the first problem you choose 4 pennies, then you have 16 coins left to pick, and these coins can be any of the 4 choices (quarters, dimes, nickels, pennies). In the second problem you choose 4 pennies, then you have 16 coins left to pick, but none of these remaining 16 coins can be pennies, thus you are picking from 3 (quarters, dimes, nickels).
 
  • #3
Oooo hah i got it!
thanks again matt!
 

1. What is the difference between "at least" and "exactly" in combinatorics?

In combinatorics, "at least" refers to the minimum number of outcomes or elements that must be present, while "exactly" refers to a specific and precise number of outcomes or elements. For example, if a question asks for at least 3 people to be chosen for a task, it means that 3 or more people can be selected. On the other hand, if the question asks for exactly 3 people to be chosen, then only 3 people can be selected.

2. How are "at least" and "exactly" used in permutations and combinations?

In permutations, "at least" is used when calculating the total number of possible arrangements or orders of a certain number of elements, while "exactly" is used when looking for a specific arrangement or order. In combinations, "at least" is used when calculating the total number of possible combinations of a certain number of elements, while "exactly" is used when looking for a specific combination.

3. Can "at least" and "exactly" be used interchangeably in combinatorial problems?

No, "at least" and "exactly" have different meanings and cannot be used interchangeably in combinatorial problems. Using the wrong term can lead to incorrect solutions and misunderstandings.

4. How do we express "at least" and "exactly" in mathematical notation?

"At least" can be expressed using the greater than or equal to symbol (≥), while "exactly" can be expressed using the equal to symbol (=). For example, if we want to express at least 5 outcomes, we can write it as n ≥ 5, where n represents the total number of outcomes. If we want to express exactly 5 outcomes, we can write it as n = 5.

5. Are there any real-world applications of "at least" and "exactly" in combinatorics?

Yes, there are many real-world applications of "at least" and "exactly" in combinatorics. For example, in probability, we often use "at least" to calculate the chances of getting a certain number of outcomes or events, while "exactly" is used to calculate the probability of getting a specific outcome. In computer science, "at least" and "exactly" are used in algorithms and data structures to determine the minimum and exact number of elements needed for certain operations. They are also commonly used in business and finance to calculate the minimum and exact number of resources or investments required for a project.

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