Question

Assertion :If ${}^n{C}_{r-1}=36{,}^n{C}_r=84$ and ${}^n{C}_{r+1}=126$, then ${}^r{C}_2=8$. Reason: ${}^n{C}_r{=}^n{C}_{n-r}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect and Reason is correct

Answer 1

The correct option is D Assertion is incorrect and Reason is correct

$\frac{{}^n{C}_{n-r}}{{}^n{C}_{n+r}}=\frac{r+1}{n-r}\Rightarrow 2n-5r=3$ ....(1)

$\frac{{}^n{C}_{r-1}}{{}^n{C}_r}=\frac{r}{n-(n-1)}=\Rightarrow 3n-10r=-3$ ....(2)

Solving (1) and (2), we get $r=3$

${}^n{C}_2{=}^3{C}_2=3$

Thus statement (1) is false and statement (2) is true.