A boy needs to select five courses from 12 available courses, out of which 5 courses are language, course. If he can choose at most 2 language courses, then the number of ways he can choose five courses is

Answer

To solve the problem of selecting 5 courses from 12 available courses, where 5 of those are language courses and the boy can choose at most 2 language courses, we can break it down into cases based on how many language courses he selects.

### Step-by-Step Solution:

1. **Identify the Total Courses and Language Courses**:
- Total courses = 12
- Language courses = 5
- Non-language courses = 12 - 5 = 7

2. **Define Cases Based on Language Course Selection**:
- Case 1: Select 0 language courses
- Case 2: Select 1 language course
- Case 3: Select 2 language courses

3. **Calculate Each Case**:

**Case 1: Selecting 0 Language Courses**
- If he selects 0 language courses, he must select all 5 from the non-language courses.
- The number of ways to choose 5 non-language courses from 7:
\[ \text{Ways} = \binom{7}{5} = 21 \]

**Case 2: Selecting 1 Language Course**
- If he selects 1 language course, he must select 4 from the non-language courses.
- The number of ways to choose 1 language course from 5 and 4 non-language courses from 7:
\[ \text{Ways} = \binom{5}{1} \times \binom{7}{4} = 5 \times 35 = 175 \]

**Case 3: Selecting 2 Language Courses**
- If he selects 2 language courses, he must select 3 from the non-language courses.
- The number of ways to choose 2 language courses from 5 and 3 non-language courses from 7:
\[ \text{Ways} = \binom{5}{2} \times \binom{7}{3} = 10 \times 35 = 350 \]

4. **Total Number of Ways**:
- Now, we add the number of ways from all cases:
\[ \text{Total Ways} = 21 + 175 + 350 = 546 \]

### Final Answer:
The number of ways the boy can choose 5 courses is **546**.

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