Question

n+1Cr+1 : nCr = 11:6 and nCr:n-1Cr-1=6:3

Answer 1

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Answer 2

Therefore the values of $r=5$ and $n=10$

Step-by-step explanation:

Given;

$_{r+1}^{n+1}\textrm{C}:_{r}^{n}\textrm{C}=11:6$  and  $_{r}^{n}\textrm{C}:_{r-1}^{n-1}\textrm{C}=6:3$

From;

$_{r+1}^{n+1}\textrm{C}:_{r}^{n}\textrm{C}=11:6$

⇒$\frac{\frac{(n+1)!}{(r+1)!(n-r)!} }{\frac{n!}{r!(n-r)!} } =11:6$

⇒$\frac{(n+1)n!r!}{(r+1)r!n!} =11:6$ (∵ $(n+1)!=(n+1)n!$)

⇒$\frac{n+1}{r+1}=11:6$

⇒$11r+11=6n+6$

⇒$6n-11r=5$____equation-1

From;

$_{r}^{n}\textrm{C}:_{r-1}^{n-1}\textrm{C}=6:3$

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

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

⇒$n=2r$

Plug the value in equation-1;

⇒$6\times 2r-11r=5$

⇒$12r-11r=5$

∴$r=5$

and $n=2\times 5=10$

So the values of $r=5$ and $n=10$