A candidate is required to answer 6 out of 10 questions divided into 2

Question

A candidate is required to answer 6 out of 10 questions divided into 2 groups each containing 5 questions. He can attempt maximum 4 questions from each group. In how many ways can he make up his choice ?

(A) 660
(B) 566
(C) 260
(D) 200
(E) 100

(A) 660
(B) 566
(C) 260
(D) 200
(E) 100

chetan2u · Answer

So two sets of 5 questions each. We are looking at ways to select these 6 questions. 
Thus, a combination question.

Two ways
1) 5C4*5C2=5*10=50.
Vice versa too is possible, so another 50.
2) 5C3*5C3=10*10=100

Total: 50+50+100=200

D

TestPrepUnlimited · Answer

We can split the two groups into group A and group B, thus the groups are independent.

We need to answer 6 questions while doing picking more than 4 from any group, so we can do 4 + 2, 3 + 3, or 2 + 4. This corresponds to :

5C4*5C2 + 5C3*5C3 + 5C2*5C4 = 5*10+10*10 + 10*5 = 200

Ans: D

ScottTargetTestPrep · Answer

Solution:
 
The candidate can answer the questions in one of the following cases:

1) 4 questions from group one and 2 questions from group two

2) 3 questions from group one and 3 questions from group two

3) 2 questions from group one and 4 questions from group two

Notice that the two groups have the same number of questions; therefore, the number of ways he can answer the questions in case 3 is the same as in case 1. Now, let’s determine the number of ways he can answer the questions in the first two cases:

Case 1:  4 questions from group one and 2 questions from group two

5C4 x 5C2 = 5 x 10 = 50

Case 2:  3 questions from group one and 3 questions from group two

5C3 x 5C3 = 10 x 10 = 100

Case 3:

(gives us the same result as Case 1), or 50

Therefore, he can answer the questions in 50 + 100 + 50 = 200 ways.

Answer: D

Sw33per · Answer

Above direct method works well, but here is the indirect method:

First, let's figure out what are the unnaceptable ways of answering the questions:
He can not answer 5 questions from 1 group, and 1 question from the other. 
Specifically, he can not take 5 questions from group A and 1 from group B, nor can he take 5 questions from group B and one question from group A.

Thus, we have:

Possible ways = Total ways - Unnaceptable ways

Possible ways = 10C6 - (5C5 x 5C1) - (5C1 x 5C5) 
Possible ways = 210 - (1 x 5) - (5 x 1)
Possible ways = 200

Answer D

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A candidate is required to answer 6 out of 10 questions divided into 2

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