The value of ${ }^{50} \mathrm{C}_4+\sum_{\mathrm{r}=1}^6{ }^{56-r} \mathrm{C}_3$ is

The value of ${ }^{50} \mathrm{C}_4+\sum_{\mathrm{r}=1}^6{ }^{56-r} \mathrm{C}_3$ is
  1. ${ }^{55} \mathrm{C}_4$
  2. ${ }^{55} \mathrm{C}_3$
  3. ${ }^{56} \mathrm{C}_3$
  4. ${ }^{56} \mathrm{C}_4$

Solution

$ \begin{aligned} & { }^{50} \mathrm{C}_4+\sum_{\mathrm{r}=1}^6{ }^{56-r} \mathrm{C}_3 \\ & \Rightarrow{ }^{50} \mathrm{C}_4+\left[{ }^{55} \mathrm{C}_3+{ }^{54} \mathrm{C}_3+{ }^{53} \mathrm{C}_3+{ }^{52} \mathrm{C}_3+{ }^{51} \mathrm{C}_3+{ }^{50} \mathrm{C}_3\right] \\ & =\left({ }^{50} \mathrm{C}_4+{ }^{50} \mathrm{C}_3\right)+{ }^{51} \mathrm{C}_3+{ }^{52} \mathrm{C}_3+{ }^{53} \mathrm{C}_3+{ }^{54} \mathrm{C}_3+{ }^{55} \mathrm{C}_3 \\ & \Rightarrow\left({ }^{51} \mathrm{C}+{ }^{51} \mathrm{C}\right)+{ }^{52} \mathrm{C}_3+{ }^{53} \mathrm{C}_3+{ }^{54} \mathrm{C}_3+{ }^{55} \mathrm{C}_3 \\ & \Rightarrow{ }^{55} \mathrm{C}_4+{ }^{55} \mathrm{C}_3={ }^{56} \mathrm{C}_4 . \end{aligned} $

Asked in: JEE Main 2005

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