(a) Of the 4 new programmers hired by a software company, 1 will work on web applications, 1 will work on desktop applications, and 2 will work on mobile applications. In how many ways can this arrangement be made?

(b) Answer the same question if 7 programmers are hired (some of them will not be assigned to any task yet).



### (a) Problem Statement: 
Out of 4 newly hired programmers, we need to assign:
- 1 to work on **web applications**
- 1 to work on **desktop applications**
- 2 to work on **mobile applications**

We can solve this in two steps:

1. **Choose the programmer for the web applications position**: There are 4 available programmers, so we can choose 1 in \( \binom{4}{1} = 4 \) ways.
2. **Choose the programmer for the desktop applications position**: After assigning 1 programmer to the web role, 3 programmers remain, so we can choose 1 in \( \binom{3}{1} = 3 \) ways.
3. **Choose the 2 programmers for mobile applications**: The 2 remaining programmers will automatically be assigned to mobile applications. There is no choice here, as they are the only ones left.

Thus, the total number of ways to assign the programmers is:
\[
4 \times 3 = 12
\]
So, there are **12 ways** to assign the 4 programmers to these roles.

---

### (b) Problem Statement:
Out of 7 newly hired programmers, we need to assign:
- 1 to work on **web applications**
- 1 to work on **desktop applications**
- 2 to work on **mobile applications**
- The remaining 3 will not be assigned to any tasks yet.

We follow similar steps to part (a):

1. **Choose the programmer for the web applications position**: There are 7 available programmers, so we can choose 1 in \( \binom{7}{1} = 7 \) ways.
2. **Choose the programmer for the desktop applications position**: After assigning 1 programmer to the web role, 6 programmers remain, so we can choose 1 in \( \binom{6}{1} = 6 \) ways.
3. **Choose 2 programmers for mobile applications**: After assigning programmers to the web and desktop roles, 5 programmers remain. We need to choose 2 out of the remaining 5 for the mobile applications. The number of ways to do this is \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \).

Thus, the total number of ways to assign the programmers is:
\[
7 \times 6 \times 10 = 420
\]
So, there are **420 ways** to assign the 7 programmers to these roles.

---

Let me know if you need more details or have any questions!

### 5 Related Questions:
1. How would the result change if more roles (e.g., cloud or AI) were added to the task list?
2. How would you calculate the number of ways if there were 2 web positions and 1 desktop position?
3. How can permutations and combinations be applied to real-world software project planning?
4. What if all 4 programmers were to work on the same type of application (say, mobile)?
5. How would the formula change if we added constraints, like specific programmers only working on certain applications?

### Tip:
When calculating combinations, remember that order does not matter. However, for assignments where roles are distinct, both combinations and multiplication principles might come into play!

(a) Of the 4 new programmers hired by a software company, 1 will work on web applications, 1 will work on desktop applications, and 2 will work on mobile applications. In how many ways can this arrangement be made? (b) Answer the same question if 7 programmers are hired (some of them will not be assigned to any task yet).

Explore how to assign 4 programmers to roles in web, desktop, and mobile applications using combinatorics. This problem involves choosing 1 programmer for web, 1 for desktop, and 2 for mobile, calculated using combinations and multiplication principles. Includes a more complex scenario with 7 programmers. Suitable for high school students in Grades 9-12.

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