Решите уравнение:[tex]5 C^{n-1}_{2n}=8 C^{n}_{2n-1} [/tex]. Created by o1l7eg17. algebra-ru

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o1l7eg17 @o1l7eg17

July 2022 1 55 Report

Решите уравнение:
[tex]5 C^{n-1}_{2n}=8 C^{n}_{2n-1} [/tex]

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Verified answer

$5C_{2n}^{~n-1}=8C_{2n-1}^{~n}\\\\\dfrac{5\cdot (2n)!}{(2n-n+1)!\cdot(n-1)!}=\dfrac{8\cdot (2n-1)!}{(2n-1-n)!\cdot n!}\\\\\dfrac{5\cdot (2n-1)!\cdot 2n}{(n+1)\cdot n!\cdot(n-1)!}=\dfrac{8\cdot (2n-1)!}{(n-1)!\cdot n!}~~~~~\Bigg|\cdot \dfrac{(n-1)!\cdot n!}{(2n-1)!}\\\\\dfrac{10n}{(n+1)}=\dfrac{8}1\\\\10n=8n+8\\2n=8\\\boxed{\boldsymbol{n=4}}$

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Проверка :

$5C_{2n}^{~n-1}=8C_{2n-1}^{~n}~;~~~~n=4\\\\5C_8^3=8C_7^4\\\\\dfrac{5\cdot 8!}{(8-3)!\cdot3!}=\dfrac{8\cdot 7!}{(7-4)!\cdot 4!}\\\\\dfrac{5\cdot 5!\cdot 6\cdot 7\cdot 8}{5!\cdot2\cdot 3}=\dfrac{8\cdot 4!\cdot 5\cdot 6\cdot 7}{2\cdot 3\cdot 4!}\\\\5\cdot 7\cdot 8=8\cdot 5\cdot 7$

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