Ms. Hernandez has 17 tomato plants that she wants to plant in rows. She will put 2 plants in some rows and 1 plant in the others. How many different ways can she plant the tomato plants?
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To solve this problem, we can use combinations. Let's consider the two options: putting 2 plants in a row or putting 1 plant in a row.
First, we need to determine the number of rows that will have 2 plants. We can have anywhere from 0 to 17 rows with 2 plants. Let's say we have "x" rows with 2 plants.
The remaining (17 - x) rows will then have 1 plant.
To find all the possible combinations, we can use the formula for combinations:
C(n, r) = n! / (r! * (n - r)!)
Where:
- n is the total number of objects (in this case, 17 tomato plants),
- r is the number of objects chosen at a time (in this case, 2 plants for each row), and
- ! represents the factorial function (the product of all positive integers less than or equal to a given positive integer).
So, the total number of ways to plant the tomato plants is the sum of all possible combinations:
Σ(C(17, x))
Where Σ denotes the sum from x = 0 to x = 17.
We can calculate this sum by adding up all the combinations for different values of x. Let's calculate them:
Σ(C(17, x)) = C(17, 0) + C(17, 1) + C(17, 2) + ... + C(17, 17)
Using the formula for combinations, we can calculate each term and add them up to find the total number of different ways Ms. Hernandez can plant the tomato plants.