The value of frac{{}^{10}C_{r}}{{}^{11}C_{r}} | vedclass

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(A) The greatest value of ${}^{n}C_{r}$ occurs at $r = \frac{n}{2}$ if $n$ is even,and at $r = \frac{n-1}{2}$ or $r = \frac{n+1}{2}$ if $n$ is odd.For

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$\frac{6}{11}$

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(A) The greatest value of ${}^{n}C_{r}$ occurs at $r = \frac{n}{2}$ if $n$ is even,and at $r = \frac{n-1}{2}$ or $r = \frac{n+1}{2}$ if $n$ is odd.For ${}^{10}C_{r}$,$n=10$ (even),so the greatest value is at $r = \frac{10}{2} = 5$.For ${}^{11}C_{r}$,$n=11$ (odd),so the greatest values are at $r = \frac{11-1}{2} = 5$ and $r = \frac{11+1}{2} = 6$.Taking $r=5$ for both,we have:$\frac{{}^{10}C_{5}}{{}^{11}C_{5}} = \frac{\frac{10!}{5!5!}}{\frac{11!}{5!6!}} = \frac{10!}{5!5!} 	imes \frac{5!6!}{11!} = \frac{10!}{11!} 	imes \frac{6!}{5!} = \frac{1}{11} 	imes 6 = \frac{6}{11}$.

The value of $\frac{{}^{10}C_{r}}{{}^{11}C_{r}}$,when both the numerator and denominator are at their greatest values,is

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