# Question: Bicycle with Optional Features

## Comment on Bicycle with Optional Features

### Is this FCP problem or

Is this FCP problem or Combinations problem.
So there are such type of questions that can be solved using both the methods?

### It can be solved using either

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.

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.

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

### I got the correct answer, but

I got the correct answer, but only because my answer came very close to 32 out of all the other answer choices. What tripped me up was the way the question was written. Judging by the language, I could not tell if 0 features was a required setup. The question should be rewritten to state (up to all 5, INCLUDING NONE).

### If I point to a table of 5

If I point to a table of 5 donuts and say that you can have ANY NUMBER of donuts (up to all 5), I think it's clear that you can choose to have zero donuts.
More importantly, there's nothing in the question that says a bicycle must have at least 1 feature

### I see your point. However,

I see your point. However, there was another question on here somewhere that specifically had "including none" on it. Just pointing out a contrast here.

Fair enough.