Question Number 221661 by mr W last updated on 09/Jun/25 Commented by mr W last updated on 09/Jun/25 $${an}\:{unsolved}\:{old}\:{question} \\ $$ Answered by MrGaster last…
Question Number 220877 by Spillover last updated on 20/May/25 Answered by Rasheed.Sindhi last updated on 20/May/25 $$\left({a}\right)\:\left({n}+\mathrm{2}\right)!+\left({n}+\mathrm{1}\right)!+{n}! \\ $$$$=\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right){n}!+\left({n}+\mathrm{1}\right){n}!+{n}! \\ $$$$={n}!\left(\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right)+\left({n}+\mathrm{1}\right)+\mathrm{1}\right) \\ $$$$\left.={n}!\left\{\:\left({n}+\mathrm{1}\right)\left({n}+\mathrm{3}\right)+\mathrm{1}\right)\right\} \\ $$$$={n}!\left({n}^{\mathrm{2}}…
Question Number 220876 by Spillover last updated on 20/May/25 Answered by Rasheed.Sindhi last updated on 20/May/25 $$\left({a}\right)\:\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right){n} \\ $$$$\:\:\:\:\:\:=\frac{\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right){n}\left({n}−\mathrm{1}\right)!}{\left({n}−\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:=\frac{\left({n}+\mathrm{2}\right)!}{\left({n}−\mathrm{1}\right)!} \\ $$ Commented by…
Question Number 220878 by Spillover last updated on 20/May/25 Answered by Rasheed.Sindhi last updated on 22/May/25 $$\frac{{n}!}{\left({n}−{r}\right)!{r}!}+\frac{\mathrm{2}×{n}!}{\left({n}−{r}+\mathrm{1}\right)!\left({r}−\mathrm{1}\right)!}+\frac{{n}!}{\left({n}−{r}+\mathrm{2}\right)!\left({r}−\mathrm{2}\right)!} \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!{r}\left({r}−\mathrm{1}\right)\left({r}−\mathrm{2}\right)!}+\frac{\mathrm{2}×{n}!}{\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!\left({r}−\mathrm{1}\right)\left({r}−\mathrm{2}\right)!}+\frac{{n}!}{\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!} \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\mathrm{1}}{{r}\left({r}−\mathrm{1}\right)}+\frac{\mathrm{2}}{\left({n}−{r}+\mathrm{1}\right)\left({r}−\mathrm{1}\right)}+\frac{\mathrm{1}}{\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)+\mathrm{2}{r}\left({n}−{r}+\mathrm{2}\right)+{r}\left({r}−\mathrm{1}\right)}{{r}\left({r}−\mathrm{1}\right)\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\left({n}−{r}\right)^{\mathrm{2}}…
Question Number 220873 by Spillover last updated on 20/May/25 Answered by mr W last updated on 20/May/25 $$\mathrm{18}!×\mathrm{19}×\mathrm{18}=\mathrm{18}×\mathrm{19}! \\ $$$${or} \\ $$$$\mathrm{20}!−\mathrm{2}×\mathrm{19}!=\mathrm{18}×\mathrm{19}! \\ $$ Commented…
Question Number 220872 by Spillover last updated on 20/May/25 Answered by mr W last updated on 21/May/25 $${P}_{{r}} ^{{n}} ={r}!×{C}_{{r}} ^{{n}} =\frac{{n}!}{\left({n}−{r}\right)!} \\ $$ Commented…
Question Number 220874 by Spillover last updated on 20/May/25 Commented by Spillover last updated on 20/May/25 Commented by Spillover last updated on 20/May/25 $${its}\:{n}!\:\:{not}\:{n} \\…
Question Number 220744 by Spillover last updated on 18/May/25 Answered by mr W last updated on 18/May/25 $${each}\:{match}\:{has}\:{three}\:{possible} \\ $$$${results} \\ $$$$\mathrm{12}\:{matches}\:{have}\:\mathrm{3}^{\mathrm{12}} =\mathrm{531}\:\mathrm{441} \\ $$$$\:{possible}\:{results}…
Question Number 220741 by Spillover last updated on 18/May/25 Answered by som(math1967) last updated on 18/May/25 $$\:\:^{\mathrm{9}} {P}_{\mathrm{4}} =\mathrm{3024} \\ $$ Commented by Spillover last…
Question Number 220740 by Spillover last updated on 18/May/25 Answered by mr W last updated on 18/May/25 $$\mathrm{52}\:{cards}:\:\mathrm{26}\:{red}\:{and}\:\mathrm{26}\:{black} \\ $$$${to}\:{select}\:\mathrm{13}\:{cards}\:{there}\:{are}\:{totally} \\ $$$${C}_{\mathrm{13}} ^{\mathrm{52}} \:{possibilities} \\…