Hints and solutions to
Make a List #5
Lonnie has a large supply of quarters, dimes, nickels, and pennies. In
how many ways could she make change for 50 cents?
-
Let's make a table, but in a systematic way -- we'll start with two
quarters, and then put more and
more coins in the other columns:
QuartersDimesNickelsPennies
2000
1210
1205
1130
1125
11110
11015
1050
1045
10310
10215
10120
10025
0500
0420
0415
04010
0340
0335
03210
03115
03020
0260
0255
02410
02315
02220
02125
02030
0180
0175
01610
01515
01420
01325
01230
01135
01040
00100
0095
00810
00715
00620
00525
00430
00335
00240
00145
00050
- Wow! Who knew there were so many ways to make change?
Looks like 49 different ways to make change for 50 cents.
- There
are many patterns in the table worth
thinking about. For example, if you're not using any quarters, there is 1
way to make the change using 5 dimes, 3 ways using 4 dimes, 5 ways using 3
dimes, and so forth until 11 ways using no dimes. And you might remember
that 1 + 3 + 5 + 7 + 9 + 11 = 62 = 36. So that counts a lot of
the ways.
- Can you find other patterns?
- Are there ways to solve this problem with less
work?
Lonnie has a large supply of quarters, dimes, nickels, and pennies. In how many ways could she make change for 50 cents?
Quarters
Dimes
Nickels
Pennies
2
0
0
0
1
2
1
0
1
2
0
5
1
1
3
0
1
1
2
5
1
1
1
10
1
1
0
15
1
0
5
0
1
0
4
5
1
0
3
10
1
0
2
15
1
0
1
20
1
0
0
25
0
5
0
0
0
4
2
0
0
4
1
5
0
4
0
10
0
3
4
0
0
3
3
5
0
3
2
10
0
3
1
15
0
3
0
20
0
2
6
0
0
2
5
5
0
2
4
10
0
2
3
15
0
2
2
20
0
2
1
25
0
2
0
30
0
1
8
0
0
1
7
5
0
1
6
10
0
1
5
15
0
1
4
20
0
1
3
25
0
1
2
30
0
1
1
35
0
1
0
40
0
0
10
0
0
0
9
5
0
0
8
10
0
0
7
15
0
0
6
20
0
0
5
25
0
0
4
30
0
0
3
35
0
0
2
40
0
0
1
45
0
0
0
50