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Q.
The sum $\displaystyle\sum_{k=0}^{n-2}{ }^n C_k \cdot{ }^n C_{k+2}$ equals
Binomial Theorem
Solution:
Write the expansion of $(1+ x )^{ n }$ in two ways, multiply and compare the co-efficient of $x ^{ n +2}$.
$ S = C _0 C _2+ C _1 C _3+ C _2 C _4+\ldots \ldots \ldots . .+ C _{ n -2} C _{ n }$
now, $(1+ x )^{ n }= C _0+ C _1 x + C _2 x ^2+\ldots \ldots \ldots \ldots . .+ C _{ n } x ^{ n } \ldots \ldots \ldots . . .(1)$
$( x +1)^{ n }= C _0 x ^{ n }+ C _1 x ^{ n -1}+ C _2 x ^{ n -2}+\ldots \ldots \ldots \ldots+ C _{ n }$.....(2)
Multiplying (1) and (2) \& equating the coefficient of $x ^{ n +2}$
$ C _0 C _2+ C _1 C _3+ C _2 C _4+\ldots \ldots \ldots \ldots \ldots . . . coefficient \text { of } x ^{ n +2} \text { in }(1+ x )^{2 n }={ }^{2 n } C _{ n +2}$