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Question
If an the expansion of \[\left( 1 + x \right)^{15}\] , the coefficients of \[\left( 2r + 3 \right)^{th}\text{ and } \left( r - 1 \right)^{th}\] terms are equal, then the value of r is
Options
5
6
4
3
Solution
5
\[\text{ Coefficients of (2r + 3)th and (r - 1)th terms in the given expansion are } ^{15}{}{C}_{2r + 2} \text{ and }^{15}{}{C}_{r - 2 .} \]
\[\text{ Thus, we have } \]
\[ ^{15}{}{C}_{2r + 2} = ^{15}{}{C}_{r - 2} \]
\[ \Rightarrow 2r + 2 = r - 2 \text{ or } 2r + 2 + r - 2 = 15 \left[ \because \text{ if } {}^n C_x =^n C_y \Rightarrow x = y \text{ or } x + y = n \right] \]
\[ \Rightarrow r = - 4 \text{ or } r = 5\]
\[\text{ Neglecting the negative value, We have } \]
\[r = 5\]
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