Category: Permutation and Combination

Question Number 66 by newuser last updated on 15/Nov/14 $$\mathrm{How}\mathrm{many}\mathrm{unique}\mathrm{arrangements}\mathrm{are}$$$$\mathrm{possible}\mathrm{for}\mathrm{a}2\times 2\mathrm{Rubic}\mathrm{cube}?$$ Commented by 123456 last updated on 14/Dec/14 $$6\times 5=30$$ Commented…

Question Number 143740 by Willson last updated on 17/Jun/21 $$\mathrm{Prove}\mathrm{that}$$$$\underset{\mathrm{n}\to +\mathrm{\infty }}{\lim 2n}-(2\mathrm{n}+1)\ln \left(\mathrm{n}\right)+\underset{\mathrm{p}=0}{\sum ^{\mathrm{n}}}\ln (1+{\mathrm{p}}^2)=\ln (e^{\pi }-e^{-\pi })$$ Answered by TheHoneyCat last updated…

Question Number 143627 by bobhans last updated on 16/Jun/21 Answered by Olaf_Thorendsen last updated on 16/Jun/21 $$\mathrm{X}=\frac{(3^x-4^x)(3^x-4.4^x)}{{\log }_2\left(6(x-\frac{1}{2})(x-\frac{4}{3})\right)}$$$${etc}… \…

Question Number 143462 by Willson last updated on 14/Jun/21 $$\underset{x\to 1}{\lim }\frac{x-1}{ln\left(\frac{x}{2-x}\right)}=???$$ Answered by Dwaipayan Shikari last updated on 14/Jun/21 $$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{{x}−\mathrm{1}}{{log}\left(\frac{{x}}{\mathrm{2}−{x}}\right)}=\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{z}}{{log}\left(\frac{\mathrm{1}+{z}}{\mathrm{1}−{z}}\right)}=\frac{{z}}{\mathrm{2}\left({z}+\frac{{z}^{\mathrm{3}} }{\mathrm{3}}+..\right)}=\frac{\mathrm{1}}{\mathrm{2}}…

Question Number 12087 by tawa last updated on 12/Apr/17 $$\mathrm{In}\mathrm{how}\mathrm{many}\mathrm{ways}\mathrm{can}10\mathrm{objects}\mathrm{be}\mathrm{split}\mathrm{into}\mathrm{two}\mathrm{groups}\mathrm{containing}$$$$4\mathrm{and}6\mathrm{objects}\mathrm{respectively}?$$ Answered by sandy_suhendra last updated on 12/Apr/17 $$10\mathrm{C}4=\frac{10!}{6!4!}=210$$$$\mathrm{or}\:\:\mathrm{10C6}\:=\:\mathrm{210} \…