Pet Parade: Mr. James has 14 cats, dogs and guinea pigs. What are all the combinations he could have?
To determine all the combinations Mr. James could have with his 14 cats, dogs, and guinea pigs, we can start by finding the different values for each type of pet combination. Let's consider the number of cats, dogs, and guinea pigs as variables:
Let x = number of cats
Let y = number of dogs
Let z = number of guinea pigs
Since Mr. James has a total of 14 pets, we can represent this as an equation:
x + y + z = 14
To find all the combinations, we need to find the integer solutions to this equation where x, y, and z are non-negative integers.
We can use a systematic approach to list all the possible combinations:
1. x = 0
In this case, the equation becomes:
0 + y + z = 14
y + z = 14
Possible solutions are:
(0, 0, 14), (0, 1, 13), (0, 2, 12), ..., (0, 13, 1), (0, 14, 0)
2. x = 1
In this case, the equation becomes:
1 + y + z = 14
y + z = 13
Possible solutions are:
(1, 0, 13), (1, 1, 12), (1, 2, 11), ..., (1, 12, 1), (1, 13, 0)
3. x = 2
In this case, the equation becomes:
2 + y + z = 14
y + z = 12
Possible solutions are:
(2, 0, 12), (2, 1, 11), (2, 2, 10), ..., (2, 10, 2), (2, 11, 1), (2, 12, 0)
Continue this process until x = 14.
Summarizing all the combinations, Mr. James could have the following combinations of pets:
(0, 0, 14), (0, 1, 13), (0, 2, 12), ..., (0, 13, 1), (0, 14, 0)
(1, 0, 13), (1, 1, 12), (1, 2, 11), ..., (1, 12, 1), (1, 13, 0)
(2, 0, 12), (2, 1, 11), (2, 2, 10), ..., (2, 10, 2), (2, 11, 1), (2, 12, 0)
...
(13, 0, 1), (13, 1, 0)
(14, 0, 0)
There are 120 total combinations.
To find all the combinations Mr. James could have with his 14 pets, we can use a combination formula.
Since Mr. James has three types of pets (cats, dogs, and guinea pigs), we can assign variables to represent the number of each type of pet. Let's use:
- Let "C" represent the number of cats.
- Let "D" represent the number of dogs.
- Let "G" represent the number of guinea pigs.
Given that Mr. James has a total of 14 pets, we know that:
C + D + G = 14
To find all the combinations, we need to consider different allocations of pets among the three categories. We can use the combination formula, which calculates the number of ways to choose a certain number of items from a larger set.
The formula for combinations is: C(n, r) = n! / (r!(n - r)!)
Where:
- C(n, r) represents the number of combinations of choosing r items from a set of n items.
- n! represents the factorial of n (the product of all positive integers less than or equal to n).
Now, let's plug in our values:
- n = 14 (the total number of pets)
- r = 3 (the number of categories)
So, we have:
C(14, 3) = 14! / (3!(14 - 3)!)
Calculating this expression will give us the number of combinations Mr. James could have with his 14 pets.
just start listing them. If the order is cats, dogs, pigs, then he could have
12 1 1
11 2 1
11 1 2
10 3 1
10 2 2
10 1 3
and so on. If you look, you may discover a pattern to keep from having to list all of them.