Final Answer:
The number of ways Ellen can assign the chores to her four kids is 480 ways.
Step-by-step explanation:
To solve this problem, we can use a combinatorial method known as "stars and bars," which is a helpful technique in combinatorics to solve problems related to distributing indistinguishable objects among distinguishable bins where each bin must contain at least one object.
Let's define the chores as "stars" (*) and the separations between kids as "bars" (|). There are 5 chores (stars) and 3 bars since Ellen has 4 kids. We want to ensure that each kid gets at least one chore.
To always give at least one chore to each child, we start by giving one chore to each:
```
* | * | * | * (Four chores have already been distributed, one for each kid)
```
That leaves us with 1 remaining chore to distribute. Now we can place this remaining chore with any of the kids. Imagine that we have 3 dividers and 1 additional chore, which we can symbolically represent as follows (where C represents a chore):
```
C | | | The remaining chore is given to the first kid.
| C | | The remaining chore is given to the second kid.
| | C | The remaining chore is given to the third kid.
| | | C The remaining chore is given to the fourth kid.
```
These additional arrangements can be placed in any of the 4 positions available, one for each kid. We use the "stars and bars" method to determine the number of ways of arranging 1 star among 4 positions (1 star + 3 bars), which can be calculated using the binomial coefficient:
```
Number of ways = (Number of stars + Number of bars) choose (Number of stars)
= (1 + 3) choose (1)
= 4 choose 1
= 4!/((4-1)!1!)
= (4*3*2*1)/((3*2*1)*1)
= 4/1
= 4
```
So there are 4 ways to distribute the remaining chore.
Now remember, the chores are distinct, and this distribution doesn't take into account the different ways to assign the specific chores. For the first four chores, which we initially assigned to ensure that each kid gets at least one, there are 5 choices for the first chore, 4 choices for the second one (since we've already given out one), 3 choices for the third, and 2 choices for the fourth.
This gives us a total of 5 * 4 * 3 * 2 ways to distribute the first four chores. Then, as calculated earlier, we have 4 ways to give out the fifth chore.
Multiplying these two results together, the total number of ways Ellen can assign the chores is:
```
5 * 4 * 3 * 2 * 4 = 480 ways
```
Therefore, there are 480 different ways for Ellen to assign the five different chores to her four kids with each kid having at least one chore.