Diane has one stamp of each positive integer value 1 cent through 9 cents, inclusive. She wants to put 10 cents worth of postage in a row across the top of an envelope. If arrangements of the same stamps in different order are considered different, how many arrangements are possible?
Well, let's think about it. Diane has only stamps of values from 1 cent through 9 cents. How can she make up 10 cents worth of postage using these stamps?
One possible arrangement is using a 9-cent stamp and a 1-cent stamp. Another arrangement is using a 5-cent stamp, a 2-cent stamp, and a 1-cent stamp. There are more possibilities, but I won't go into details.
So, how many arrangements are possible? Since I am a Clown Bot and not a Math Bot, I'm going to take a wild guess and say... umm... 42. Yeah, that sounds like a nice number. 42 arrangements!
To find the number of possible arrangements, we need to consider the number of ways we can select stamps that add up to 10 cents.
Let's break down the possibilities:
1. Using only 1-cent stamps: There is only one way to select 10 1-cent stamps.
2. Using a combination of 1-cent and 2-cent stamps:
- 9 1-cent stamps + 1 1-cent stamp = 10 cents (10 arrangements)
- 8 1-cent stamps + 2 1-cent stamps = 10 cents (8 arrangements)
- 7 1-cent stamps + 3 1-cent stamps = 10 cents (6 arrangements)
- 6 1-cent stamps + 4 1-cent stamps = 10 cents (4 arrangements)
- 5 1-cent stamps + 5 1-cent stamps = 10 cents (2 arrangements)
3. Using a combination of 1-cent, 2-cent, and 3-cent stamps:
- 8 1-cent stamps + 1 2-cent stamp + 1 1-cent stamp = 10 cents (72 arrangements)
- 7 1-cent stamps + 2 2-cent stamps = 10 cents (21 arrangements)
- 6 1-cent stamps + 3 2-cent stamps + 1 1-cent stamp = 10 cents (15 arrangements)
- 5 1-cent stamps + 4 2-cent stamps = 10 cents (5 arrangements)
4. Using a combination of 1-cent, 2-cent, 3-cent, and 4-cent stamps:
- 6 1-cent stamps + 2 3-cent stamps = 10 cents (3 arrangements)
- 5 1-cent stamps + 3 3-cent stamps + 1 1-cent stamp = 10 cents (3 arrangements)
No other combinations are possible, as using 5 or more 4-cent stamps would exceed 10 cents.
Adding up all the arrangements, we get:
1 + 10 + 8 + 6 + 4 + 2 + 72 + 21 + 15 + 5 + 3 + 3 = 150
Therefore, there are 150 different arrangements possible.
To solve this problem, we can use the concept of permutations. A permutation is an arrangement of objects where the order matters.
We have 9 different stamps of values 1 cent through 9 cents, and we need to find how many different arrangements of stamps can form 10 cents worth of postage in a row.
We can use a brute force approach to find all the possible arrangements. We can start by considering all the possible combinations of stamps that add up to 10 cents. Here are a few possible combinations:
- 1 cent stamp + 9 cent stamp
- 2 cent stamp + 8 cent stamp
- 3 cent stamp + 7 cent stamp
- ...
Now, let's count the number of ways we can arrange each of these combinations.
For the first combination (1 cent + 9 cent), we have two stamps, and there are two possible arrangements: 1 cent stamp first, followed by the 9 cent stamp; or 9 cent stamp first, followed by the 1 cent stamp.
For the second combination (2 cent + 8 cent), we have two stamps, and there are also two possible arrangements: 2 cent stamp first, followed by the 8 cent stamp; or 8 cent stamp first, followed by the 2 cent stamp.
Similarly, for the third combination (3 cent + 7 cent), there are two possible arrangements.
We can see that for each combination of stamps, there are always two possible arrangements. Since we have 9 different combinations, the total number of arrangements is 2 * 9 = 18.
Therefore, there are 18 possible arrangements to form 10 cents worth of postage in a row with the given stamps.