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Question

A question paper on mathematics consists of twelve questions divided into three parts $$A,B$$ and $$C$$ each containing four questions. In how many ways can an examinee answer five questions, selecting atleast one from each part.


A
624
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B
208
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C
2304
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D
None
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Solution

The correct option is A $$624$$
Five questions can be answered in the following way

 $$A$$$$B$$$$C$$  
 $$1$$$$1$$$$3$$ $$4C_1 \times 4C_1 \times 4C_3$$ $$64$$
 $$1$$$$2$$ $$2$$  $$4C_1 \times 4C_1 \times 4C_3$$ $$144$$
 $$2$$$$2$$$$1$$ $$4C_1 \times 4C_1 \times 4C_3$$  $$144$$
 $$2$$ $$1$$ $$2$$ $$4C_1 \times 4C_1 \times 4C_3$$ $$144$$
 $$3$$ $$1$$ $$1$$ $$4C_1 \times 4C_1 \times 4C_3$$ $$64$$
 $$1$$ $$3$$ $$1$$ $$4C_1 \times 4C_1 \times 4C_3$$ $$64$$
Therefore total number of ways $$=64+144+144+144+64+64=624$$


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