In how many ways can a students choose 5 courses out of 9 courses if 2 courses are compulsory for every students?

Answer

To solve the problem of how many ways a student can choose 5 courses out of 9 courses when 2 courses are compulsory, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Courses and Compulsory Courses**:
- There are a total of 9 courses available.
- Out of these, 2 courses are compulsory.

2. **Choose the Compulsory Courses**:
- Since the 2 courses are compulsory, there is only 1 way to choose these 2 courses (you must take them).

\[ \text{Ways to choose compulsory courses} = \binom{2}{2} = 1 \]

3. **Determine Remaining Courses**:
- After selecting the 2 compulsory courses, we have \(9 - 2 = 7\) courses left.

4. **Choose the Remaining Courses**:
- The student needs to choose 3 more courses from the remaining 7 courses (since 5 total courses are needed and 2 are already chosen).
- The number of ways to choose 3 courses from 7 is given by the combination formula:

\[ \text{Ways to choose remaining courses} = \binom{7}{3} \]

5. **Calculate the Combination**:
- Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), we calculate \(\binom{7}{3}\):

\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!} \]

- Simplifying this:

\[ = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

6. **Total Number of Ways**:
- Since there is 1 way to choose the compulsory courses and 35 ways to choose the remaining courses, the total number of ways to choose the courses is:

\[ \text{Total ways} = 1 \times 35 = 35 \]

### Final Answer:
The total number of ways a student can choose 5 courses out of 9 courses, with 2 courses being compulsory, is **35**.

---