You received a $100 gift certificate to a clothing store. The store sells t-shirts for $15 and dress shirts for $22. You want to spend no more than the amount of the gift certificate. You want to leave at most $10 of the gift certificate unspent. You need at least one dress shirt What are all of the possible combinations of t-shirts and dress shirts you can buy?

To find all the possible combinations of t-shirts and dress shirts you can buy, follow these steps:

Step 1: Determine the maximum number of dress shirts you can buy
Since you want to spend no more than the gift certificate amount and leave at most $10 unspent, subtracting $10 from $100 gives you $90 to spend. Divide this by the price of a dress shirt ($22) to find the maximum number of dress shirts you can buy: $90 ÷ $22 = 4 dress shirts.

Step 2: Generate combinations
Start by considering each possible number of dress shirts and its corresponding number of t-shirts. Then, calculate the remaining balance for each combination. If the remaining balance is between $0 and $10, it is a valid combination.

Here are all the possible combinations:

Combination 1: 4 dress shirts and 0 t-shirts
Total cost = (4 * $22) + (0 * $15) = $88 + $0 = $88
Remaining balance = $100 - $88 = $12 (valid combination)

Combination 2: 3 dress shirts and 2 t-shirts
Total cost = (3 * $22) + (2 * $15) = $66 + $30 = $96
Remaining balance = $100 - $96 = $4 (valid combination)

Combination 3: 2 dress shirts and 4 t-shirts
Total cost = (2 * $22) + (4 * $15) = $44 + $60 = $104
Remaining balance = $100 - $104 = -$4 (invalid combination)

Combination 4: 1 dress shirt and 6 t-shirts
Total cost = (1 * $22) + (6 * $15) = $22 + $90 = $112
Remaining balance: $100 - $112 = -$12 (invalid combination)

Combination 5: 0 dress shirts and 8 t-shirts
Total cost = (0 * $22) + (8 * $15) = $0 + $120 = $120
Remaining balance: $100 - $120 = -$20 (invalid combination)

Now you have two valid combinations:
1) 4 dress shirts and 0 t-shirts
2) 3 dress shirts and 2 t-shirts

To find all the possible combinations of t-shirts and dress shirts that fulfill your requirements, we can use a systematic approach. Let's break it down step by step:

1. Determine the limit for each type of shirt:
- T-shirts: Since they cost $15 each, you can buy a maximum of ($100 - $10) / $15 = $90 / $15 = 6 t-shirts.
- Dress shirts: Since you need to buy at least one and they cost $22 each, you can buy a maximum of ($100 - $22) / $22 = $78 / $22 = 3 dress shirts.

2. Create a table to organize the combinations:

T-shirts | Dress shirts
---------|--------------
0 | 1
0 | 2
0 | 3
1 | 1
1 | 2
1 | 3
2 | 1
2 | 2
2 | 3
3 | 1
3 | 2
3 | 3
4 | 1
4 | 2
4 | 3
5 | 1
5 | 2
5 | 3
6 | 1
6 | 2
6 | 3

3. Calculate the total cost for each combination:
- For each row in the table, multiply the number of t-shirts by $15 and the number of dress shirts by $22. Then add them up to get the total cost for that combination.

4. Eliminate the combinations that exceed the gift certificate amount:
- Remove any combination whose total cost is greater than $100.

By following these steps, you'll obtain all the possible combinations that satisfy the given conditions.

If you spend $x, you want

90 <= x <= 100
so, if you buy m T-shirts and n dress shirts, you need
15m+22n = x
n >= 1
So, start checking:
with n=1, 15m+22 = x
n m x
1 5 22+15*1=97
2 3 44+15*3=89; 44+15*4=104 so 2 dress shirts will not work.
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