Q. A candidate is required to answer $6$ out of $10$ questions, which are divided into two groups, each containing $5$ questions. He is not permitted to attempt more than $4$ questions from either group. The number of different ways in which the candidate can choose $6$ questions is
Permutations and Combinations
Solution:
The number of ways the candidate can choose questions under the given conditions is enumerated below.
Group 1
Group 2
Number of ways
4
2
$(^5C_4)(^4C_2) = 50$
3
3
$(^5C_3)(^5C_3)= 100$
5
4
$(^5C_2)(^5C_4) = 50$
Total number of ways
200
| Group 1 | Group 2 | Number of ways |
|---|---|---|
| 4 | 2 | $(^5C_4)(^4C_2) = 50$ |
| 3 | 3 | $(^5C_3)(^5C_3)= 100$ |
| 5 | 4 | $(^5C_2)(^5C_4) = 50$ |
| Total number of ways | 200 |