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
B
4
C
6
D
8

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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