JEE MAIN - Mathematics (2023 - 24th January Morning Shift - No. 18)
A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is __________
Answer
546
Explanation
Among 12 courses, 5 courses are of language.
$$\therefore$$ Remaining 7 are different courses.
Now, number of ways to select 5 courses where at most 2 language courses present.
| Language | Different | Number of ways | |
|---|---|---|---|
| Case 1 | 0 | 5 | $${}^5{C_0} \times {}^7{C_5}$$ |
| Case 2 | 1 | 4 | $${}^5{C_1} \times {}^7{C_4}$$ |
| Case 3 | 2 | 3 | $${}^5{C_2} \times {}^7{C_3}$$ |
$$\therefore$$ Total number of ways
$$ = {}^5{C_0} \times {}^7{C_5} + {}^5{C_1} \times {}^7{C_4} + {}^5{C_2} \times {}^7{C_3}$$
$$ = 546$$
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