Question

Given that C(n, r) : C ( n, r + 1)=1 : 2 and C (n, r + 1) : C (n, r + 2) = 2 : 3.

Find $r.$

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The correct option is C 4

Given : $\frac{{}_r^{n}C}{{}_{r+1}^{n}C}=\frac{1}{2}$

$\Rightarrow 2\frac{n!}{r!(n-r)!}=\frac{n!}{(r+1)!(n-r-1)!}$

$\Rightarrow 3r+2=n$

Also, $\frac{{}_{r+1}^{n}C}{{}_{r+2}^{n}C}=\frac{2}{3}$

$\Rightarrow 3\frac{n!}{(r+1)!(n-r-1)!}=2\frac{n!}{(r+2)!(n-r-2)!}$

$\Rightarrow 3r+6=2n-2r-2$

$\Rightarrow 5r+8=2n$

Substituting $n$, we get

$8+5r=2(3r+2)$

$8+5r=6r+4$

$\Rightarrow r=4$

Hence, C is correct.