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Assertion (A): The number of selections of 20 distinct things taken 8 at a time is same as that taken 12 at a time.
Reason (R): $\mathrm{C}(\mathrm{n}, \mathrm{r})=\mathrm{C}(\mathrm{n}, \mathrm{s})$, if $\mathrm{n}=\mathrm{r}+\mathrm{s}$
Options:
Reason (R): $\mathrm{C}(\mathrm{n}, \mathrm{r})=\mathrm{C}(\mathrm{n}, \mathrm{s})$, if $\mathrm{n}=\mathrm{r}+\mathrm{s}$
Solution:
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Verified Answer
The correct answer is:
Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
Number of selection of 20 distinct things taken 8 at a time is given by
${ }^{20} \mathrm{C}_{8}=\frac{20 !}{12 ! 8 !}$
and selecting 12 out of 20 is
${ }^{20} \mathrm{C}_{12}=\frac{20 !}{12 ! 8 !}$
Thus, both ${ }^{20} \mathrm{C}_{8}$ and ${ }^{20} \mathrm{C}_{12}$ are same.
$\Rightarrow$ Both $\mathrm{A}$ and $\mathrm{R}$ are individually true and $\mathrm{R}$ is correct explanation of A.
${ }^{20} \mathrm{C}_{8}=\frac{20 !}{12 ! 8 !}$
and selecting 12 out of 20 is
${ }^{20} \mathrm{C}_{12}=\frac{20 !}{12 ! 8 !}$
Thus, both ${ }^{20} \mathrm{C}_{8}$ and ${ }^{20} \mathrm{C}_{12}$ are same.
$\Rightarrow$ Both $\mathrm{A}$ and $\mathrm{R}$ are individually true and $\mathrm{R}$ is correct explanation of A.
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