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Question

A question paper is divided into two parts $$A$$ and $$B$$ and each part contains $$5$$ questions. The number of ways in which a candidate can answer $$6$$ questions selecting at least two questions from each part is:


A
80
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B
100
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C
200
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D
None of these
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Solution

The correct option is C $$200$$
 $$ (A)$$ $$(B)$$ 
 $$ (5\, question)$$$$(5\, question)$$  
$$ (i)$$$$ 2$$$$ 4$$$$ =6$$
$$(ii)$$$$ 3$$$$ 3$$$$ =6$$
$$( iii)$$$$ 4$$$$ 2$$$$ =6$$
There are two section and the candidates has to answer at least $$2$$ question from each section with total of $$6$$ questions.
So, there are three followign ways.
$$(i)$$ no of ways $$\Rightarrow $$ selecting $$2$$ questions from section $$A$$ and $$4$$ questions from sections $$B$$
$$\Rightarrow ^5C_2\times ^5C_4 \Rightarrow 10\times 5 \Rightarrow 50$$
$$(ii)$$ no. of ways $$\Rightarrow $$ selecting $$3$$ questions from each section.
$$\Rightarrow  ^5C_3\times ^5C_3\Rightarrow 10\times 19=100$$
$$(iii)$$ no. of ways $$\Rightarrow $$ selecting $$4$$ from section $$A$$ and $$2$$ from section $$B$$ 
$$\Rightarrow ^5C_4\times ^5C_2\Rightarrow 5 \times 10\Rightarrow 50$$
$$\therefore $$ Total no. of ways $$=50+100+50=200$$. 

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