If ${ }^{11} \mathrm{C}_4+{ }^{11} \mathrm{C}_5+{ }^{12} \mathrm{C}_6+{ }^{13} \mathrm{C}_7={ }^{14} \mathrm{C}_{\mathrm{r}}$, then value of $\mathrm{r}$ is

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Aicharya · Accepted Answer

${ }^{11} \mathrm{C}_4+{ }^{11} \mathrm{C}_5+{ }^{12} \mathrm{C}_6+{ }^{13} \mathrm{C}_7={ }^{14} \mathrm{C}_{\mathrm{r}}$ We know that ${ }^n C_r+{ }^n C_{r-1}={ }^{n+1} C_r$ $\begin{aligned} & 	herefore{ }^{14} \mathrm{C}_{\mathrm{r}}=\left({ }^{11} \mathrm{C}_4+{ }^{11} \mathrm{C}_5ight)+{ }^{12} \mathrm{C}_6+{ }^{13} \mathrm{C}_7 \ & =\left({ }^{12} \mathrm{C}_5+{ }^{12} \mathrm{C}_6ight)+{ }^{13} \mathrm{C}_7 \ & ={ }^{13} \mathrm{C}_6+{ }^{13} \mathrm{C}_7={ }^{14} \mathrm{C}_7 \ & 	herefore \mathrm{r}=7 \end{aligned}$

If ${ }^{11} \mathrm{C}_4+{ }^{11} \mathrm{C}_5+{ }^{12} \mathrm{C}_6+{ }^{13} \mathrm{C}_7={ }^{14}…

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