Find r if ^ 14 C_ 2 r : ^ 10 C_ 2 r-4 =143: 10

Question

Find $r$ if ${ }^{14} \mathrm{C}_{2 \mathrm{r}}:{ }^{10} \mathrm{C}_{2 \mathrm{r}-4}=143: 10$

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Answer

$
\begin{aligned}
& { }^{14} \mathrm{C}_{2 \mathrm{r}}:{ }^{10} \mathrm{C}_{2 \mathrm{r}-4}=143: 10 \\
& \therefore \quad \frac{14 !}{2 r !(14-2 r) !} \div \frac{10 !}{(2 r-4) !(14-2 r) !}=\frac{143}{10} \\
& \therefore \quad \frac{14 !}{2 \mathrm{r} !(14-2 \mathrm{r}) !} \times \frac{(2 \mathrm{r}-4) !(14-2 \mathrm{r}) !}{10 !}=\frac{143}{10} \\
& \therefore \quad \frac{14 \times 13 \times 12 \times 11 \times 10 !}{2 r(2 r-1)(2 r-2)(2 r-3)(2 r-4) !(14-2 r) !} \\
& \times \frac{(2 r-4) !(14-2 r) !}{10 !}=\frac{143}{10} \\
& \\
& \therefore \frac{14 \times 13 \times 12 \times 11}{2 \mathrm{r}(2 \mathrm{r}-1) \times(2 \mathrm{r}-2)(2 \mathrm{r}-3)}=\frac{143}{10} \\
& \therefore 2 \mathrm{r}(2 \mathrm{r}-1)(2 \mathrm{r}-2)(2 \mathrm{r}-3)=14 \times 12 \times 10 \\
& \therefore 2 \mathrm{r}(2 \mathrm{r}-1)(2 \mathrm{r}-2)(2 \mathrm{r}-3)=8 \times 7 \times 6 \times 5 \\
& \text { Comparing on both sides, we get } \\
& \therefore r=4 \\
&
\end{aligned}
$

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