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Question

Find the value of $$r$$, if coefficient of $$(2r+4)^{th}$$ and $$(r-2)^{th}$$ terms in expansion of $$(1+x)^{18}$$ are equal:


A
5
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B
4
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C
6
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D
8
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Solution

The correct option is C $$6$$
We know that in the coefficient of $$r^{th}$$ term in expansion of $$(1+x)^n={}^{n}C_{r-1}$$
$$\therefore$$ coefficients of $$(2r+4)^{th}$$ and $$(r-2)^{th}$$ in the 
expansion $$(1+x)^{18}$$ are $${}^{18}C_{2r+3}$$ and $${}^{18}C_{r-3}.$$

Now, 

$${}^{18}C_{2r+3}={}^{18}C_{r-3}$$

 $$2r+3=r-3$$   or, $$2r+3+r-3=18$$   $$(\because {}^n C_r={}^n C s$$ then $$r=s$$ or $$r+s=n)$$

On solving,

 $$2r+3=r-3$$

 $$r=-6$$ which is not possible.

$$2r+3+r-3=18$$

 $$3r=18$$

 $$r=6$$

Hence, C is the correct option.

Mathematics

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