If
\[
{}^nC_{r-1}=36,\qquad {}^nC_r=84,\qquad {}^nC_{r+1}=126,
\]
then the value of \(nr^2\) is
Show Hint
For consecutive binomial coefficients, use
\[
\frac{{}^nC_r}{{}^nC_{r-1}}
=
\frac{n-r+1}{r}
\]
and
\[
\frac{{}^nC_{r+1}}{{}^nC_r}
=
\frac{n-r}{r+1}.
\]
These formulas quickly determine \(n\) and \(r\).
Step 1: Form two ratio equations. \(\frac{^nC_r}{^nC_{r-1}}=\frac{n-r+1}{r}=\frac{84}{36}=\ frac{7}{3}\Rightarrow 3n-10r+3=0\). \(\frac{^nC_{r+1}}{^nC_r}=\frac{n-r}{r+1}=\frac{126}{84}=\frac{3}{2}\Rightarrow 2n-5r-3=0\).
Step 2: Solve the two linear equations. From (i): \(3n=10r-3\); from (ii): \(2n=5r+3\). Multiply (ii) by 2: \(4n=10r+6\). Subtract (i): \(n=9\), then \(r=3\). So \(nr^2=9\times9=81\). \[ \boxed{81} \]