Solve-C-x-y-C-x-y-1-4C-x-y-5C-x-y-1-or-C-n-k-n-k-n-k-

Question Number 170833 by LEKOUMA last updated on 01/Jun/22

S o l v e

{\begin{cases} C_{x}^{y} = C_{x}^{y + 1} \\ 4 C_{x}^{y} = 5 C_{x}^{y - 1} \end{cases} o r C_{n}^{k} = \frac{n!}{k! (n - k)!}

Answered by aleks041103 last updated on 04/Jun/22

C_{x}^{y} = C_{x}^{y + 1}

\frac{x!}{y! (x - y)!} = \frac{x!}{(y + 1)! (x - y - 1)!}

\frac{(y + 1)!}{y!} = \frac{(x - y)!}{(x - y - 1)!}

y + 1 = x - y

x - 2 y = 1 \Rightarrow x = 1 + 2 y

4 C_{x}^{y} = 5 C_{x}^{y - 1}

4 \frac{x!}{y! (x - y)!} = 5 \frac{x!}{(y - 1)! (x - y + 1)!}

4 \frac{(x - y + 1)!}{(x - y)!} = 5 \frac{y!}{(y - 1)!}

4 (x - y + 1) = 5 y

9 y - 4 x = 1

\Rightarrow 9 y - 4 (1 + 2 y) = 1

9 y - 4 - 8 y = 1

y = 5 \Rightarrow x = 11