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Question Number 41824 by Tawa1 last updated on 13/Aug/18
$$\mathrm{if}\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{5}} \:=\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{11}} \:\:\:\mathrm{find}\:\:\:\:\:\:\overset{\mathrm{18}} {\:}\mathrm{C}_{\mathrm{n}} \\ $$
Answered by $@ty@m last updated on 13/Aug/18
$$\:\:^{{n}} {C}_{{r}} =\:^{{n}} {C}_{{n}−{r}} \\ $$$$\Rightarrow{r}=\mathrm{5} \\ $$$$\&\:{n}−{r}=\mathrm{11}\Rightarrow{n}=\mathrm{16} \\ $$$$\Rightarrow\:^{\mathrm{18}} {C}_{{n}} =\:^{\mathrm{18}} {C}_{\mathrm{16}} =\:^{\mathrm{18}} {C}_{\mathrm{2}} =\mathrm{153} \\ $$
Commented by Tawa1 last updated on 13/Aug/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 13/Aug/18
$$\frac{{n}×\left({n}−\mathrm{1}\right)×\left({n}−\mathrm{3}\right)×\left({n}−\mathrm{4}\right)}{\mathrm{5}!}=\frac{{n}×\left({n}−\mathrm{1}\right)...×\left({n}−\mathrm{10}\right)}{\mathrm{11}!} \\ $$$$\left({n}−\mathrm{5}\right)\left({n}−\mathrm{6}\right)\left({n}−\mathrm{7}\right)\left({n}−\mathrm{8}\right)\left({n}−\mathrm{9}\right)\left({n}−\mathrm{10}\right)=\mathrm{11}×\mathrm{10}×\mathrm{9}×\mathrm{8}×\mathrm{7}×\mathrm{6} \\ $$$${n}=\mathrm{16} \\ $$$$\mathrm{18}{c}_{\mathrm{16}} =\frac{\mathrm{18}×\mathrm{17}}{\mathrm{2}}=\mathrm{153} \\ $$
Commented by Tawa1 last updated on 13/Aug/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Answered by malwaan last updated on 13/Aug/18
$$\mathrm{n}=\mathrm{5}+\mathrm{11}=\mathrm{16} \\ $$$$\therefore\:\underset{\mathrm{n}} {\overset{\mathrm{18}} {\mathrm{C}}}=\underset{\mathrm{16}} {\overset{\mathrm{18}} {\mathrm{C}}}=\frac{\mathrm{18}×\mathrm{17}}{\mathrm{2}×\mathrm{1}}=\mathrm{9}×\mathrm{17} \\ $$$$\:\:\:=\mathrm{153} \\ $$
Commented by Tawa1 last updated on 13/Aug/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$