Exploring the Possibilities of Coin Combinations: Counting Without Making Change

[Alexsandro]

This question seems easy, but I don't understand what it's mean with "without making change". Could someone help me?

If you have 2 dollars, 2 quarter and 3 nickels:

a) how many different sums can you pay without making change?

b) Change the quarters into dimes and answer again.
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This question is in a book and the answers are:
a)35 (0 sum being excluded);
b)23
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To the item (a), I did this way:
if it can to be used [0,1 or 2] dollars, [0,1 or 2] quarters and [0,1,2, or 3] nickels, then there are 3x3x4 = 36 - 1 = 35 different sums, excluding the 0 sum.
Is my reasoning correct?
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I don't find the answer to item (b). Could you help me ?

 

[NateTG]

The tricky part for (b) is, of course, that 2 Nickels = 1 Dime. Perhaps you can find a different, easier set of coins that will generate the same sums as 2 dollars, 2 dimes, and 3 nickels?

 

[Alexsandro]

I didn't understand yet

The tricky part for (b) is, of course, that 2 Nickels = 1 Dime. Perhaps you can find a different, easier set of coins that will generate the same sums as 2 dollars, 2 dimes, and 3 nickels?

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I am trying, but I still don't obtaining the answer to part (b). Do you know how to do this ?

 

[EnumaElish]

In this context "without making change" means:

1. using up all the coins and getting no money back from the seller,

and

2. not exchanging any coins with coins of the same aggregate value but of different individual values. (e.g. you are not allowed to replace 1 Real with four 0.25 Reals.)

 

[EnumaElish]

Part (b) is asking "how would your answer to part (a) be modified if we took away the two 0.25 Reals and gave you five 0.10 Reals?" (Now you have 2 dollars, 5 dimes and 3 nickels.)

 

[EnumaElish]

To the item (a), I did this way:
if it can to be used [0,1 or 2] dollars, [0,1 or 2] quarters and [0,1,2, or 3] nickels, then there are 3x3x4 = 36 - 1 = 35 different sums, excluding the 0 sum.
Is my reasoning correct?

Seems correct to me.

 

[NateTG]

Your problem is that the nickels and dimes intersect, right?

Let's try an easier version:

How many different sums can you make with 7 nickels?
How many different sums can you make with 3 dimes and 1 nickel?

 

[balakrishnan_v]

Using 2 dimes and 3 nickels,all the amonts within 1 dollar thar can be paid are
0c,5c,10c,15c,20c,25c,30c,35c which is 8
So when combined with $ it is
8*3-1(exclude 0$)=23

 

[Alexsandro]

Thanks

Thanks, I understood and I could response this question!