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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)2 Marks
Question
Find $\mathrm{n}$ if ${ }^{21} \mathrm{C}_{6 \mathrm{n}}={ }^{21} C_{n^2+5}$
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Answer

$
\begin{aligned}
& { }^{21} C_{6 n}={ }^{21} C_{n^2+5} \\
& \text { If }{ }^n C_x={ }^n C_y \text {, then either } x=y \text { or } x=n-y \\
& \therefore 6 n=n^2+5 \text { or } 6 n=21-\left(n^2+5\right) \\
& \therefore n^2-6 n+5=0 \text { or } 6 n=21-n^2-5 \\
& \therefore n^2-6 n+5=0 \text { or } n^2+6 n-16=0 \\
& \text { If } n^2-6 n+5=0 \text { then }(n-1)(n-5)=0 \\
& \therefore n=1 \text { or } n=5 \\
& \text { If } n=5 \text { then } n^2+5=30>21 \\
& \therefore n \neq 5 \\
& \therefore n=1 \\
& \text { If } n^2+6 n-16=0 \text { then }(n+8)(n-2)=0 \\
& n=-8 \text { or } n=2 \\
& n \neq-8 \\
& \therefore n=2
\end{aligned}
$

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