Ms. Hernandez has 17 tomato plants that she wants to plant in rows. She will put 1 plant in some rows and 2 plants in the others. How many different ways can she plant the tomato plants? Make a table to solve.
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To find the number of different ways Ms. Hernandez can plant the tomato plants, we can create a table to examine all possible combinations of rows with 1 plant and rows with 2 plants.
Let's start by considering the number of rows with 2 plants. Since there can be at most 17 tomato plants and each row with 2 plants requires two plants, the maximum number of rows with 2 plants can be 17 / 2 = 8 rows.
Now, let's construct a table to represent the different arrangements of rows with 2 plants. For simplicity, we'll use the letter 'A' to represent rows with 2 plants and the letter 'B' to represent rows with 1 plant.
Number of rows with 2 plants (A) | Number of rows with 1 plant (B) | Total rows
----------------------------------|----------------------------------|-------------
1 | 16 | 17
2 | 15 | 17
3 | 14 | 17
4 | 13 | 17
5 | 12 | 17
6 | 11 | 17
7 | 10 | 17
8 | 9 | 17
In this table, the number of rows with 2 plants and the number of rows with 1 plant, when added together, always equal 17 (the total number of rows). We list out all the possible combinations, starting from 1 row with 2 plants and reducing it as we increase the number of rows with 1 plant.
Now let's calculate the number of different ways to arrange the tomato plants. For each row with 2 plants, there can be 2 tomato plants arranged within that row. Similarly, for each row with 1 plant, there can be 1 tomato plant arranged within that row.
Therefore, the total number of different ways to arrange the tomato plants can be calculated by summing up the number of ways to arrange the tomato plants within each row (2^(number of rows with 2 plants) * 1^(number of rows with 1 plant)).
Let's calculate the different ways for each combination in the table:
Number of rows with 2 plants (A) | Number of rows with 1 plant (B) | Total rows | Different ways
----------------------------------|----------------------------------|-----------------|------------------
1 | 16 | 17 | 2^1 * 1^16 = 2
2 | 15 | 17 | 2^2 * 1^15 = 4
3 | 14 | 17 | 2^3 * 1^14 = 8
4 | 13 | 17 | 2^4 * 1^13 = 16
5 | 12 | 17 | 2^5 * 1^12 = 32
6 | 11 | 17 | 2^6 * 1^11 = 64
7 | 10 | 17 | 2^7 * 1^10 = 128
8 | 9 | 17 | 2^8 * 1^9 = 256
By summing up all the different ways for each combination in the table, we find that there are a total of 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 = 510 different ways Ms. Hernandez can plant the tomato plants.