Question 719395
<pre>
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:

&#8373;   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


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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:

 &#8373;  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:

 &#8373;  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:

 &#8373;  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</pre>