Question

If ncr:ncr+1:ncr+2are in ratio with 1:2:3, then find the values of n and r.

Answer 1

n=3r+1 and r=6 is the best answer

Answer 2

n = 14 and r = 4

Step-by-step explanation:

Given that $^nC_{r} : ^nC_{r + 1} : ^nC_{r + 2} = 1 : 2 : 3$

So, $\frac{^nC_{r}}{^nC_{r + 1}} = \frac{1}{2}$

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

⇒ $\frac{r + 1}{n - r} = \frac{1}{2}$

⇒ n - 3r = 2 ........ (1)

⇒ 2n - 6r = 4 ........ (2)

Again, we have $\frac{^nC_{r + 1}}{^nC_{r + 2}} = \frac{2}{3}$

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

⇒ $\frac{r + 2}{n - r - 1} = \frac{2}{3}$

⇒ 2n - 5r = 8 ......... (3)

Now, solving equations (2) and (3) we get,

r = 4 and from equation (1) we get, n = 2 + 3r = 14

Therefore, n = 14 and r = 4 (Answer)