Question

A vending machine dispensing books of stamps accepts only one-dollar coins, $1 bills, and $5 bills.

How many ways are there to deposit $10 for a book of stamps?

Note: I have the answer but do not know how to solve for it. Answer: 1,217

Answer 1

make 3 columns
$5 1C 1B ----- where 1C is 1$ coin and 1B is 1$ bill

2 0 0
1 5 0
1 4 1
1 3 2
1 2 3
1 1 4
1 0 5
0 10 0
0 9 1
0 8 2
0 7 3
0 6 4
0 5 5
0 4 6
0 3 7
0 2 8
0 1 9
0 0 10

count them up, I don't see any others to make $10

I can't see where that large answer comes from, unless I misinterpret the question.

Answer 2

We will assume "ways" means order is important, i.e. $5 followed by 5 $1 coins is not the same "way" as 5 $1 coins.

We will separate the problem into a few parts.
A. No $5 bill
$1 bills and coins only, ten of them.
Each $1 we have a choice of two ways, coin or bill, so there are 2^10=1024 ways.
B. One $5 bill.
There are 2^5=32 ways to put the dollars, and 6 positions to put the $5 bill among the 5 dollars, so that makes 6*32=192 ways.
C. Two $5 bills
There is only one way.

Total = 1024+192+1=1217 ways.

Answer 3

Thank you for the help MathMate I really appreciate it :) And yes in this question order matters. I apologize this was a 3 part question and that was mentioned in part a). I was able to understand all parts but the last one. Your explanation makes sense.