SOLUTION: Australian coins are made in 6 denominations: 1₵, 2₵, 5₵, 10₵, 20₵, 50₵ and $1. Your tasks are to determine all the different ways a pers

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Question 719395: Australian coins are made in 6 denominations: 1₵, 2₵, 5₵, 10₵, 20₵, 50₵ and $1.
Your tasks are to determine all the different ways a person could make from those coins total amounts of:
7₵
10₵
20₵ (answer here is >35)
30₵ (here without using any 1₵ coins)
For each of the 4 cases, all the ways must be shown in a logically sequenced list and you must carefully explain in words the logical sequencing “rules” you used so the reader can see that your list is complete.
P.S. The summation of all the ways across all 4 cases will approach 75.
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
To make 7-cents, we can use either 0 or 1 5-cent coin.

If we use 0 5-cent coin, we can use from 0 to 3, inclusive, 2-cent coins
and the rest in 1-cent coins.  That's 4 ways.

If we use 1 5-cent coin, we can get the remaining 2 cents by either 1
2-cent coin, or 2 1-cent coins.  That's 2 more ways.

Total number of ways for having 7-cents is 4+2 or 6 ways.

Here are the 6 ways:

₵   5   2   1
-------------     
1.  0   0   7
2.  0   1   5
3.  0   2   3
4.  0   3   1
5.  1   0   2
6.  1   1   0


---------------
 
To make 10-cents, we can use either 0 or 1 10-cent coin.

I.   If we use 0 10-cent coin, then we can use 0,1, or 2 5-cent coins:
     A.  If we use 0 5-cent coins, then we can use 0 through 5 2-cent coins,
         inclusive, 2-cent coins and the rest, if any, in 1-cent coins.  
         That's 6 ways.
     B.  If we use 1 5-cent coin, then we can use 0 through 2, inclusive,
         2-cent coins and the rest, if any, in 1-cent coins.  That's 3 ways.
     C.  If we use 2 5-cent coins, that's 10 cents, so that's 1 way.

II.  If we use 1 10-cent coin, that's 1 way.

Total number of ways for having 10-cents is 6+3+1+1 or 11 ways. 

Here are the 11 ways:

 ₵  10   5   2   1
------------------- 
 1.  0   0   0  10
 2.  0   0   1   8
 3.  0   0   2   6
 4.  0   0   3   4
 5.  0   0   4   2
 6.  0   0   5   0
 7.  0   1   0   5
 8.  0   1   1   3
 9.  0   1   2   1
10.  0   2   0   0
11.  1   0   0   0
  

-----------------------

To make 20-cents, we can use either 0 or 1 20-cent coin.

I.   Use 0 20-cent coin, then we can use 0,1,or 2 10-cent coins.
     A. Use 0 10-cent coins, then we can use 0,1,2,3, or 4 5-cent coins.   
        1. 0 5-cent coins. then we can use 0 through 10, inclusive,
           2-cent coins, and the rest, if any, in 1-cent coins.  
           That's 11 ways.
        2. 1 5-cent coins. then we have 15 cents left to make. Then we 
           can use 0 through 7, inclusive, 2-cent coins, and the rest
           in 1-cent coins.  That's 8 ways.
        3. 2 5-cent coins. then we have 10 cents left to make. Then we 
           can use 0 through 5, inclusive, 2-cent coins, and the rest
           in 1-cent coins.  That's 6 ways.
        4. 3 5-cent coins. then we have 5 cents left to make. Then we 
           can use 0 through 2, inclusive, 2-cent coins, and the rest
           in 1-cent coins.  That's 3 ways.
        5. 4 5-cent coins.  That's 20 cents. That's 1 way.
     B. Use 1 10-cent coin.  Then to make the remaining 10-cents, this
        is the same as the preceding problem (of making 10 cents) minus
        the one case where we used exactly 1 10-cent coin to make the 10
        cents.  That's 11-1 or 10 ways.
     C. Use 2 10-cent coins.  That's 20 cents.  That's 1 way.
II.  1 20-cent coin.  That's 1 way.

Total number of ways for having 20-cents is 11+8+6+3+1+10+1+1 or 41 ways.

Here are the 42 ways:

 ₵  20  10   5   2   1
----------------------
 1.  0   0   0   0  20
 2.  0   0   0   1  18
 3.  0   0   0   2  16
 4.  0   0   0   3  14
 5.  0   0   0   4  12
 6.  0   0   0   5  10
 7.  0   0   0   6   8
 8.  0   0   0   7   6
 9.  0   0   0   8   4
10.  0   0   0   9   2
11.  0   0   0  10   0
12.  0   0   1   0  15
13.  0   0   1   1  13
14.  0   0   1   2  11
15.  0   0   1   3   9
16.  0   0   1   4   7
17.  0   0   1   5   5
18.  0   0   1   6   3
19.  0   0   1   7   1
20.  0   0   2   0  10
21.  0   0   2   1   8
22.  0   0   2   2   6
23.  0   0   2   3   4
24.  0   0   2   4   2
25.  0   0   2   5   0
26.  0   0   3   0   5
27.  0   0   3   1   3
28.  0   0   3   2   1
29.  0   0   4   0   0
30.  0   1   0   0  10
31.  0   1   0   1   8
32.  0   1   0   2   6
33.  0   1   0   3   4
34.  0   1   0   4   2
35.  0   1   0   5   0
36.  0   1   1   0   5
37.  0   1   1   1   3
38.  0   1   1   2   1
39.  0   1   2   0   0
40.  0   2   0   0   0
41.  1   0   0   0   0

-----------------------

To make 30-cents, we can use either 0 or 1 20-cent coins.  Since we aren't
using any 1-cent coins, we can only use an even number of 5-cent coins.

I.   If we use 0 20-cent coin, then we can use 0,1,2 or 3 10-cent coins:
     A.  If we use 0 10-cent coins, then we can use 0,2,4 or 6 2-cent coins,
         inclusive, 2-cent coins and the rest, if any, in 2-cent coins.  
         That's 4 ways.
     B.  If we use 1 10-cent coin, then we can use 0,2, or 4 5-cent coins and
         the rest, if any, in 2-cent coins.  That's 3 ways.
     C.  If we use 2 10-cent coins, then we can use 0 or 2 5-cent coins and
         the rest, if any, in 2-cent coins.  That's 2 ways.
     D.  If we use 3 10-cent coins, that's 30 cents.  That's 1 way.

II.  If we use 1 20-cent coin, then we can use 0 or 1 10-cent coin
     A. If we use 0 10-cent coins, then we can use 0 or 2 5-cent
        coins, and the rest, if any, in 2-cent coins. That's 2 ways.
     B. If we use 1 10-cent coin, then that's 30 cents. That's 1 way.

Total number of ways for having 30-cents is 4+3+2+1+2+1 or 13 ways. 

Here are the 13 ways:

 ₵  20  10   5   2 
------------------
 1.  0   0   0  15   
 2.  0   0   2  10   
 3.  0   0   4   5   
 4.  0   0   6   0   
 5.  0   1   0  10   
 6.  0   1   2   5   
 7.  0   1   4   0   
 8.  0   2   0   5   
 9.  0   2   2   0   
10.  0   3   0   0   
11.  1   0   0   5   
12.  1   0   2   0   
13.  1   1   0   0   

Edwin
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