Муодиларо ҳал намоед:
\(C_{x}^{x-1}+C_{x}^{x-2}+C_{x}^{x-3}+...+C_{x}^{x-8}+C_{x}^{x-9}+C_{x}^{x-10}=1023\)
A\(=C_{x}^{x-1}+C_{x}^{x-2}+C_{x}^{x-3}+...+C_{x}^{x-8}+C_{x}^{x-9}+C_{x}^{x-10}\)
\(C_{n}^{n-k}=C_{n}^{k}\)
A\(=C_{x}^{1}+C_{x}^{2}+C_{x}^{3}+...+C_{x}^{8}+C_{x}^{9}+C_{x}^{10}\)
\(C_{x}^{1}+C_{x}^{2}+C_{x}^{3}+...+C_{x}^{8}+C_{x}^{9}+C_{x}^{10}=1023\,\,\,\,\,|+1\)
\(1+C_{x}^{1}+C_{x}^{2}+C_{x}^{3}+...+C_{x}^{8}+C_{x}^{9}+C_{x}^{10}=1024\)
\(C_{x}^{0}=1\)
\(C_{x}^{0}+C_{x}^{1}+C_{x}^{2}+C_{x}^{3}+...+C_{x}^{8}+C_{x}^{9}+C_{x}^{10}=1024\)
\(C_{x}^{0}+C_{x}^{1}+C_{x}^{2}+C_{x}^{3}+...+C_{x}^{8}+C_{x}^{9}+C_{x}^{10}=2^{10}\)
\(C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+C_{n}^{3}+...+C_{n}^{n-2}+C_{n}^{n-1}+C_{n}^{n}=2^{n}\)
\(C_{x}^{0}+C_{x}^{1}+C_{x}^{2}+C_{x}^{3}+...+C_{x}^{8}+C_{x}^{9}+C_{x}^{10}=2^{10}\)
\(\Rightarrow{x = 10}\)
Муодила ҳал шуд.
Ҷавоб: x = 10