Looking for formula for combo's of coins totalling 25 cents

erikam

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How many combinations of pennies, nickels, dimes, and quarters can you put together that equal $0.25?

The teacher told my daughter to make a list (yes, that sounds easy), but that isn't an option for other combination questions - especially when you may have a 100 combinations depending on the question. Is there a formula for this?
 
How many combinations of pennies, nickels, dimes, and quarters can you put together that equal $0.25?

The teacher told my daughter to make a list (yes, that sounds easy), but that isn't an option for other combination questions - especially when you may have a 100 combinations depending on the question. Is there a formula for this?
No, there isn't a direct formula for answering the exercise that you've posted.

I don't know your daughter's class or grade level, and I can't know the teacher's motivation, but perhaps having students list the possibilities is for assessing organization skills and/or introducing some abstraction (i.e., pre-algebra). Listing sounds easy, but some students will not get all thirteen possibilities. Sometimes listing stuff on paper is a good approach. :cool:



There is a symbolic way to represent the value of the coin combinations.

combination amount (in cents) = 25*a + 10*b + 5*c + d

where a,b,c,d are symbols representing numbers of coins (and the asterisks are multiplication signs).

This sort of thing is done in beginning algebra. For example, that equation could be used with other equations to solve an exercise like: 300 coins are on the floor -- quarters, dimes, nickels, pennies -- and their total value is $28.50. There are ten pennies for every nickel. If the quarters and nickels were separated from the dimes and pennies, the pile with dimes and pennies would contain twice as many coins. How many quarters, dimes, nickels, and pennies are there?

In intermediate algebra (or pre-calculus), some classes will discuss counting principles, combinations and permutations. There are various algorithms, notations, and formulas for those. This topic would apply to an exercise like: In a lottery, six ping-pong balls are randomly chosen from a set of balls numbered 1 through 52. How many different combinations are possible?
 
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How many combinations of pennies, nickels, dimes, and quarters can you put together that equal $0.25?

The teacher told my daughter to make a list (yes, that sounds easy), but that isn't an option for other combination questions - especially when you may have a 100 combinations depending on the question. Is there a formula for this?
I agree with mmm that this does not appear to be a question soluble by formula, but it does submit to organized thought.

Possibilities

How many ways can you choose 1 type of coin out of 4 types? 4. There is a formula for this, but the answer is obvious.

All pennies: 1
All nickels: 1.
All dimes: 0.
All quarters: 1. (After this we can ignore the quarter.)

How many ways can you choose 2 types of coin out of the 3 remaining types (we can now ignore the quarter)? The answer is 3. There is a formula, but you can easily work it out without it. But this doesn't end the analysis because there may be several ways to pick nickels and pennies to add up to 25 cents.

Pennies and nickels: 4 (We could have 1, 2, 3, or 4 nickels.)
Pennies and dimes: 2.
Nickels and dimes: 2.

How many ways can we choose 3 types of coin out of 3 types. Obviously 1 (although there is a formula). Again though we must consider how many different ways we can combine pennies, nickels, and dimes to get 25 cents.


1 dime, pennies and nickels: 2.
2 dimes, pennies, and nickels: 0. (With 2 dimes, we can get to 25 cents with 1 nickel, but then there are no pennies.)

So I get 13 ways.
 
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