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2C-4-n-35C-3-n-2-n-




Question Number 74129 by malwaan last updated on 19/Nov/19
2C_4 ^n  = 35C_3 ^(n/2)    ⇒ n = ?
$$\mathrm{2}\boldsymbol{{C}}_{\mathrm{4}} ^{\boldsymbol{{n}}} \:=\:\mathrm{35}\boldsymbol{{C}}_{\mathrm{3}} ^{\frac{\boldsymbol{{n}}}{\mathrm{2}}} \: \\ $$$$\Rightarrow\:\boldsymbol{{n}}\:=\:? \\ $$
Answered by MJS last updated on 20/Nov/19
n∈N ⇒ n≥4∧(n/2)≥3 ⇒ n≥6  n=2k (k≥3)  2C_4 ^(2k) =35C_3 ^k   2(((2k)!)/(4!(2k−4)!))=35((k!)/(3!(k−3)!))  (1/3)k(k−1)(2k−3)(2k−1)=((35)/6)k(k−2)(k−1)  k=0∨k=1 not valid  (1/3)(2k−3)(2k−1)=((35)/6)(k−2)  2(2k−3)(2k−1)=35(k−2)  8k^2 −51k+76=0  (k−4)(8k−19)=0  k=((19)/8) not valid  k=4  ⇒ n=8
$${n}\in\mathbb{N}\:\Rightarrow\:{n}\geqslant\mathrm{4}\wedge\frac{{n}}{\mathrm{2}}\geqslant\mathrm{3}\:\Rightarrow\:{n}\geqslant\mathrm{6} \\ $$$${n}=\mathrm{2}{k}\:\left({k}\geqslant\mathrm{3}\right) \\ $$$$\mathrm{2}{C}_{\mathrm{4}} ^{\mathrm{2}{k}} =\mathrm{35}{C}_{\mathrm{3}} ^{{k}} \\ $$$$\mathrm{2}\frac{\left(\mathrm{2}{k}\right)!}{\mathrm{4}!\left(\mathrm{2}{k}−\mathrm{4}\right)!}=\mathrm{35}\frac{{k}!}{\mathrm{3}!\left({k}−\mathrm{3}\right)!} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}{k}\left({k}−\mathrm{1}\right)\left(\mathrm{2}{k}−\mathrm{3}\right)\left(\mathrm{2}{k}−\mathrm{1}\right)=\frac{\mathrm{35}}{\mathrm{6}}{k}\left({k}−\mathrm{2}\right)\left({k}−\mathrm{1}\right) \\ $$$${k}=\mathrm{0}\vee{k}=\mathrm{1}\:\mathrm{not}\:\mathrm{valid} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{2}{k}−\mathrm{3}\right)\left(\mathrm{2}{k}−\mathrm{1}\right)=\frac{\mathrm{35}}{\mathrm{6}}\left({k}−\mathrm{2}\right) \\ $$$$\mathrm{2}\left(\mathrm{2}{k}−\mathrm{3}\right)\left(\mathrm{2}{k}−\mathrm{1}\right)=\mathrm{35}\left({k}−\mathrm{2}\right) \\ $$$$\mathrm{8}{k}^{\mathrm{2}} −\mathrm{51}{k}+\mathrm{76}=\mathrm{0} \\ $$$$\left({k}−\mathrm{4}\right)\left(\mathrm{8}{k}−\mathrm{19}\right)=\mathrm{0} \\ $$$${k}=\frac{\mathrm{19}}{\mathrm{8}}\:\mathrm{not}\:\mathrm{valid} \\ $$$${k}=\mathrm{4} \\ $$$$\Rightarrow\:{n}=\mathrm{8} \\ $$
Commented by malwaan last updated on 20/Nov/19
thank you so much sir MJS
$${thank}\:{you}\:{so}\:{much}\:{sir}\:{MJS} \\ $$
Commented by MJS last updated on 20/Nov/19
you′re welcome
$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$

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