Permutations and Combinations 2 Question 25
25. If ${ }^{n} C _{r-1}=36,{ }^{n} C _r=84$ and ${ }^{n} C _{r+1}=126$, then find the values of $n$ and $r$.
$(1979,3$ M)
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Solution:
- We know that, $\frac{{ }^{n} C _r}{{ }^{n} C _{r-1}}=\frac{n-r+1}{r}$
$$ \begin{array}{ll} \Rightarrow & \frac{84}{36}=\frac{7}{3}=\frac{n-r+1}{r} \\ \Rightarrow & 3 n-10 r+3=0 \end{array} $$
Also given, $\quad \frac{{ }^{n} C _r}{{ }^{n} C _{r+1}}=\frac{84}{126}$
$$ \begin{aligned} \Rightarrow & \frac{r+1}{n-r} & =\frac{2}{3} \\ \Rightarrow & 2 n-5 r-3 & =0 \end{aligned} $$
On solving Eqs. (i) and (ii), we get
$$ r=3 \quad \text { and } \quad n=9 $$
