Question

A candidate to answer 6 out of 10 questions which is divided into two groups A and B each containing 5 questions. He is not permitted to attempt more than 4 questions from either group. Find the number of different ways in which the candidate can choose six questions.

Answer 1

Hint: Use combinations for the two different groups and add them.
First, we have to find the total number of combinations that are totally possible and then since we are given restriction in every group, we are going to find three different conditions without violating the conditions. Then the obtained combinations we are going to add them and find the required combinations.

Complete step by step solution:
First, find the total number of combinations we can get.
${}^{10}{C_6} = 210$${}^{10}{C_6} = 210$
To solve the given problem, we have two approaches
First Method
We are going to subtract the possibility of choosing more than 4 question from either group from the total possible outcome
\[Combination = 2 \times {}^5{C_5} \times {}^5{C_1} = 10\,\,ways\]
Now, we should subtract it from the total outcomes.
The required number of ways =$210 - 10 = 200\,\,ways$
Second Method
In this method we are going to find different combinations from the pair of the given groups

Ways of selecting 3 questions from group A and 3 questions from group B=${}^5{C_3} \times {}^5{C_3}$ ${}^5{C_3} \times {}^5{C_3}$
Ways of selecting 4 question from group A and 2 questions from group B=${}^5{C_4} \times {}^5{C_2}$
Ways of selecting 2 question from group A and 4 questions from group B= ${}^5{C_2} \times {}^5{C_4}$
We will add all the total combinations to get the required ways
The required number of ways =${}^5{C_3} \times {}^5{C_3} + {}^5{C_4} \times {}^5{C_2} + {}^5{C_2} \times {}^5{C_4}$
$
   = 10 \times 10 + 5 \times 10 + 10 \times 5 \\
   = 100 + 50 + 50 \\
   = 200 \\
$
Note: They must note that both cases are different even if the values are the same because groups A and B are different and they will contain different questions. Also, take care that in every way, both conditions that are given in questions are satisfied.