You have a team of 5 people and 3 tasks to assign. Task A needs two people, Task B needs two people, and Task C needs one person. You are only going to assign one task to each person. In how many different ways can you assign the tasks?

Answer

To solve the problem of assigning tasks to a team of 5 people, we will follow these steps:

### Step 1: Select 2 people for Task A
We need to choose 2 people from the 5 available. The number of ways to choose 2 people from 5 can be calculated using the combination formula:

\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} \]

Calculating this gives:

\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 2: Select 2 people for Task B
After assigning 2 people to Task A, we have 3 people left. Now, we need to select 2 people for Task B from these remaining 3. The number of ways to choose 2 people from 3 is:

\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2! \cdot 1!} \]

Calculating this gives:

\[ \binom{3}{2} = \frac{3}{1} = 3 \]

### Step 3: Assign the remaining person to Task C
After assigning 2 people to Task A and 2 people to Task B, there will be 1 person left. This person will automatically be assigned to Task C. The number of ways to choose 1 person from 1 is:

\[ \binom{1}{1} = 1 \]

### Step 4: Calculate the total number of ways to assign tasks
Now, we can find the total number of ways to assign the tasks by multiplying the number of ways to assign each task:

\[ \text{Total Ways} = \binom{5}{2} \times \binom{3}{2} \times \binom{1}{1} = 10 \times 3 \times 1 = 30 \]

Thus, the total number of different ways to assign the tasks is **30**.

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