Twelve new clients need to be assigned to 3 lawyers. If each lawyer must get at least 3 clients, then
how many ways can the clients be assigned?

To determine the number of ways to assign 12 new clients to 3 lawyers such that each lawyer gets at least 3 clients, we can use the "stars and bars" combinatorial method combined with the inclusion-exclusion principle.

### Step 1: Express the Problem in Terms of Combinatorics

We need to solve the equation:
\[
x_1 + x_2 + x_3 = 12
\]
where \(x_1\), \(x_2\), and \(x_3\) are the number of clients assigned to the three lawyers. Each \(x_i\) must satisfy \(x_i \geq 3\) since each lawyer must get at least 3 clients.

### Step 2: Transform the Variables

We can perform a substitution to simplify the problem. Let:
\[
y_1 = x_1 - 3, \quad y_2 = x_2 - 3, \quad y_3 = x_3 - 3
\]
This transforms the equation to:
\[
y_1 + y_2 + y_3 = 3
\]
where \(y_1, y_2, y_3 \geq 0\). Now, we need to find the number of non-negative integer solutions to this equation.

### Step 3: Apply the Stars and Bars Theorem

The number of non-negative integer solutions to the equation \(y_1 + y_2 + y_3 = 3\) is given by the binomial coefficient:
\[
\binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} = 10
\]

### Step 4: Account for the Different Arrangements

The number of ways to assign clients to the lawyers, considering the lawyers are distinguishable, is given by:
\[
10 \times \binom{12}{3, 3, 6} 
\]

However, \(\binom{12}{3, 3, 6}\) represents the number of ways to choose clients where one lawyer gets 6 clients, which does not respect the rule that all \(x_i\) must be greater than or equal to 3.

### Step 5: Count the Total Assignments

Given that we have 10 different distributions of clients among the three lawyers after ensuring that each lawyer gets at least 3 clients, and there are no other constraints, the total number of ways the clients can be assigned (without the lawyers' permutation) is:

\[
\text{Total number of ways} = 10
\]

Hence, the number of ways to assign the clients such that each lawyer gets at least 3 clients is **10**.

### Final Answer:

The total number of ways to assign 12 clients to 3 lawyers such that each lawyer gets at least 3 clients is \( \boxed{220} \).

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Would you like to see any detailed explanations, or do you have any questions?

Here are some related questions:
1. How do we calculate the number of combinations when assigning items with specific conditions?
2. What is the stars and bars method in combinatorics?
3. Can you explain the inclusion-exclusion principle?
4. What would happen if the condition "at least 3 clients" were not imposed?
5. How do we handle assignments where the number of items is larger or smaller than the number required for each group?
6. What if the lawyers were indistinguishable? How would the calculation change?
7. How would the answer change if there were more than three lawyers or fewer clients?
8. What is the significance of the binomial coefficient in combinatorial problems?

**Tip:** When dealing with combinatorial problems involving conditions, breaking down the problem into smaller parts and using transformations can simplify the calculations.

Explore the combinatorial method for assigning 12 clients to 3 lawyers, ensuring each lawyer receives at least 3 clients. This detailed solution uses the stars and bars method combined with the inclusion-exclusion principle to determine the number of ways to assign clients. Suitable for advanced high school students and beyond.

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