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

Question

Find $r$ if ${ }^{14} C_{2 r}:{ }^{10} C_{2 r-4}=143: 10$.

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Answer

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

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