Toddrick has a photo album. Some pages have one photo on them, and other pages have two photos on them. If Toddrick has 9 photos, how many different ways can he put in the album?
3 ways
3 ways
3 Ways
To find the number of different ways Toddrick can arrange the photos in his album, we can consider the different possibilities for the number of pages with one photo and the number of pages with two photos.
Let's denote the number of pages with one photo as x and the number of pages with two photos as y. Since the total number of pages in the album is not given, we will consider all possible scenarios and then determine the number of ways.
We know that the total number of photos is 9. Each page can either have one or two photos, so we have the equation:
x + 2y = 9
To find the number of solutions, we need to find the non-negative integer solutions for x and y.
One way to approach this is to start with a fixed value for y and solve for x. Then, increment y and find additional solutions until all possibilities are exhausted.
Let's do this step by step:
1. Let's start with y = 0. Substituting this into the equation, we get x + 2(0) = 9.
Solving for x, we find x = 9. So, when y = 0, there is one solution: (9, 0).
2. Now, let's assume y = 1. Substituting into the equation, we have x + 2(1) = 9.
Solving for x, we find x = 7. So, when y = 1, there is one solution: (7, 1).
3. Next, we assume y = 2. Substituting into the equation, we have x + 2(2) = 9.
Solving for x, we get x = 5. Thus, when y = 2, there is one solution: (5, 2).
4. We continue this process until the equation no longer yields a positive integer value for x.
Next, we assume y = 3. Substituting, we have x + 2(3) = 9.
Solving for x, we find x = 3. When y = 3, there is one solution: (3, 3).
5. We can continue with y = 4. Substituting, x + 2(4) = 9.
Solving for x, we get x = 1. When y = 4, there is one solution: (1, 4).
At this point, we can observe that further incrementing y will result in negative values for x, which we do not consider.
Therefore, we have found all the possible solutions:
(9, 0), (7, 1), (5, 2), (3, 3), (1, 4)
Each solution represents a different way of arranging the photos in the album. So, there are a total of 5 different ways that Toddrick can put the photos in his album.