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Permutations and Combinations (p-2) — Maths (commerce) STD 11 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 11 Commerce / ArtsMaths (commerce)Permutations and Combinations (p-2)1 Mark
Question
Find $n$ if ${ }^{2 n} C_{r-1}={ }^{2 n} C_{r+1}$
✓

Answer

$
\begin{aligned}
& { }^{2 n} C_{r-1}={ }^{2 n} C_{r+1} \\
& \text { If }{ }^n C_x={ }^n C_y \text {, then either } x=y \text { or } x=n-y \\
& \therefore r-1=r+1 \text { or } r-1=2 n-(r+1)
\end{aligned}
$
But $r-1=r+1$ is not possible
$
\begin{aligned}
& \therefore \mathrm{r}-1=2 \mathrm{n}-(\mathrm{r}+1) \\
& \therefore \mathrm{r}+\mathrm{r}=2 \mathrm{n} \\
& \therefore \mathrm{r}=\mathrm{n}
\end{aligned}
$

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