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  4. If 1+ displaystyle∑ r =1 n n + r C r = 25 C 13, n ∈ N, then the value of n equals

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Q. If $1+\displaystyle\sum_{ r =1}^{ n }{ }^{ n + r } C _{ r }={ }^{25} C _{13}, n \in N$, then the value of $n$ equals

Binomial Theorem

Solution:

$1+{ }^{n+1} C_1+{ }^{n+2} C_2+\ldots . .+{ }^{n+n} C_n={ }^{25} C_{13} \Rightarrow{ }^{1+2 n} C_n={ }^{25} C_{13} \Rightarrow n=12$