The least value of the natural number $n$ satisfying $C(n, 5)+C(n, 6)>C(n+1,5)$ is

The least value of the natural number $n$ satisfying $C(n, 5)+C(n, 6)>C(n+1,5)$ is
  1. $10$
  2. $11$
  3. $12$
  4. $13$

Solution

$\begin{aligned} & \text { We have, }{ }^n C_5+{ }^n C_6>{ }^{n+1} C_5 \\ & { }^{n+1} C_6>{ }^{n+1} C_5 \\ & \Rightarrow \quad \frac{(n+1) !}{6 !(n+1-6) !}>\frac{(n+1) !}{5 !(n+1-5) !} \\ & \Rightarrow \quad 5 !(n-4) !>6 !(n-5) ! \\ & \Rightarrow \quad 5 !(n-4)(n-5) !>6 \cdot 5 !(n-5) ! \\ & \Rightarrow \quad n-4>6 \\ & \Rightarrow \quad n>10 \\ & \Rightarrow \quad n=11 \\ & \end{aligned}$

Asked in: AP EAMCET 2002

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