Question: Bicycle with Optional Features

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Is this FCP problem or Combinations problem.
So there are such type of questions that can be solved using both the methods?
greenlight-admin's picture

It can be solved using either the Fundamental Counting Principle (FCP) or combinations.

Both approaches are discussed in the video.

From 0:45 to 1:35, I use the FCP approach.

From 1:35 to 3:05, I use combinations.

Could you please help me out with this sum
In an entrance test a candidate is required to attempt a total of four questions which are to be attempted from 2 section each containing 5 questions.The maximum number of questions that he can attempt from any section is 3.In how many ways can he answer in the test.
For the above example I tried doing 5C3×5C1 + 5C2x5C2 + 5C1x5C3.
I wanted to know whether it is correct to consider 5C0x5C4 also.
Please help!
greenlight-admin's picture

Your solution (5C3×5C1 + 5C2x5C2 + 5C1x5C3) already considers all possible cases.

That is, if we call one section Section A and call the other section Section B, then there are 3 possible cases:

ANSWER 3 QUESTIONS FROM SECTION A AND 1 QUESTION FROM SECTION B
Number of possible outcomes = 5C3 x 5C1

ANSWER 2 QUESTIONS FROM SECTION A AND 2 QUESTIONS FROM SECTION B
Number of possible outcomes = 5C2 x 5C2

ANSWER 1 QUESTION FROM SECTION A AND 3 QUESTIONS FROM SECTION B
Number of possible outcomes = 5C1 x 5C3

So, TOTAL outcomes = 5C3×5C1 + 5C2x5C2 + 5C1x5C3

The last value you suggested (5C0x5C4) represents answering 0 questions from Section A and answering 4 questions from Section B. This breaks the rule that says "The maximum number of questions that he can attempt from any section is 3." So, we must not include this in our final answer.

Does that help?

Cheers,
Brent

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