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$$ 500!\times499! =500\times499!\times499!$$ $$= 500\times(499!)^2.$$ $$ 500!\times499!\times498!\times\dots\times1! = 500\times498\times\dots\times2\times(499!\times497!\times\dots\times1!)^2 $$ $$ = (2\times250)\times(2\times249)\times\dots\times(2\times1)\times(499!\times497!\times\dots\times1!)^2 $$ $$ = 2^{250}\times250!\times(499!\times497!\times\dots\times1!)^2 $$

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$$\frac{n!}{n-123}-\frac{123}{n-123}$$

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$$1\times1!+2\times2!+...+k\times k!+(k+1)\times(k+1)!$$$$=(k+1)!-1+(k+1)\times(k+1)!$$ $$=(1+k+1)(k+1)!-1$$ $$=(k+2)!-1$$

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$$1!\times2!\times3!\times...\times100! =1^{100}\times2^{99}\times3^{98}\times...\times100^1$$ $$=\left[1^{100}\times3^{98}\times...\times99^2\right]\times\left[2^{99}\times4^{97}\times...\times100^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times\left[2^{99}\times4^{97}\times...\times100^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times\left[(2\times1)^{99}\times(2\times2)^{97}\times...\times(2\times50)^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times2^{99+97+...+1}\left[1^{99}\times2^{97}\times...\times50^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times2^{2500}\left[1^{99}\times2^{97}\times...\times50^1\right]$$ $$\frac{1!\times2!\times3!\times...\times100! }{50!}=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times2^{2500}\left[1^{98}\times2^{96}\times...\times50^0\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\times2^{1250}\times1^{49}\times2^{48}\times...\times50^0\right]^2$$

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