19. In how many ways can 20 kopecks be exchanged for coins worth 5 kopecks, 2 kopecks, and 1 kopeck?
19. Let's list all the ways to change 20 kopecks. In such a change, we can use either four 5-kopeck coins, or three such coins, or two, or one, or none.

There is only one way to change 20 kopecks using four 5-kopeck coins (since four 5-kopeck coins already make 20 kopecks).

If we use three 5-kopeck coins, we need to make up another 5 kopecks with coins of other denominations. This can be done in three ways: take two 2-kopeck coins and one 1-kopeck coin, or one 2-kopeck coin and three 1-kopeck coins, or, finally, no 2-kopeck coins and five 1-kopeck coins.

Thus, we have indicated three more ways to change 20 kopecks.

If we use two 5-kopeck coins, we need to make up another 10 kopecks with 2-kopeck and 1-kopeck coins. This can be done in six ways. We can take:

1) 5 coins of 2 kopecks, 0 coins of 1 kopeck.
2) 4 » 2 » 2 » » 1 »
3) 3 » " 2 » 4 » » 1 »
4) 2 » » 2 » 6 » » 1 »
5) 1 » " 2 » 8 » » 1 »
6) 0 » " 2 » 10 » " 1 » 20 kopecks.

Thus, we have indicated six more ways to change 20 kopecks.

If we use one 5-kopeck coin, we need to make up another 15 kopecks with coins of other denominations. This can be done in the following ways:

1) 7 coins of 2 kopecks, 1 coin of 1 kopeck.
2) 6 " " 2 » 3 » » 1 »

![](https://cdn.mathpix.com/cropped/2024_05_21_68334347b64402497ba5g-117.jpg?height=236&width=689&top_left_y=1375&top_left_x=198)

Thus, we have indicated eight more ways to change 20 kopecks.

Finally, if 5-kopeck coins are not used at all in the change, the following ways are possible:

| 1) 10 | coins | of | 2 kopecks, |  | coins |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | » | » | 2 " | 2 | " | » | 1 |
| 3) | » | » | 2 » | 4 | " | » | 1 |
| 7 | " | » | 2 » | 6 | » | » | 1 |
| 6 | » | » | 2 » | 8 | » | » | 1 |
| 5 | » | » | 2 » | 10 | » | » | 1 |
| 4 | » | » | 2 " | 12 | » | » | 1 |
| 3 | » | » | 2 " | 14 | » | » | 1 |
| 2 | » | » | 2 " | 16 | » | " | 1 |
| 1 | " | » | 2 " | 18 | » | " | 1 |
| n | » | ” | 2 " | 20 | » | " | 1 |

i.e., 11 more ways.

All the indicated ways of changing are different, and there are no other ways. Therefore, the total number of ways to change 20 kopecks is

$$
1+3+6+8+11=29
$$