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Category: Permutation and Combination

Show-that-n-C-r-n-n-1-n-2-n-r-1-r-

Question Number 150554 by pete last updated on 13/Aug/21 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{n}_{\mathrm{C}_{\mathrm{r}} } =\frac{\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)\left(\mathrm{n}−\mathrm{2}\right)…\left(\mathrm{n}−\mathrm{r}+\mathrm{1}\right)}{\mathrm{r}!} \\ $$ Answered by Ar Brandon last updated on 13/Aug/21 $$\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)…\left({n}−{r}+\mathrm{1}\right)}{{r}!} \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)…\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!}{\left({n}−{r}\right)!{r}!}…

100-apples-should-be-packed-in-three-boxes-and-each-box-should-contain-at-least-10-apples-in-how-many-ways-can-this-be-done-

Question Number 84993 by mr W last updated on 18/Mar/20 $$\mathrm{100}\:{apples}\:{should}\:{be}\:{packed}\:{in}\:{three} \\ $$$${boxes}\:{and}\:{each}\:{box}\:{should}\:{contain} \\ $$$${at}\:{least}\:\mathrm{10}\:{apples}.\:{in}\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}? \\ $$ Commented by john santu last updated…

If-a-group-consist-of-8-men-and-6-women-in-how-many-ways-can-a-committee-of-5-be-selected-if-i-the-committee-is-to-consist-of-3-men-and-3-women-ii-there-are-no-restrictions-on-the-nu

Question Number 149955 by jlewis last updated on 08/Aug/21 $$\mathrm{If}\:\mathrm{a}\:\mathrm{group}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{8}\:\mathrm{men}\:\mathrm{and}\:\mathrm{6}\:\mathrm{women}, \\ $$$$\mathrm{in}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{a}\:\mathrm{committee}\:\mathrm{of} \\ $$$$\mathrm{5}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{if}: \\ $$$$\left.\:\:\:\:\:\:\mathrm{i}\right)\:\mathrm{the}\:\mathrm{committee}\:\mathrm{is}\:\mathrm{to}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{3}\:\mathrm{men} \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{women}. \\ $$$$\left.\:\:\:\:\:\:\mathrm{ii}\right)\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{restrictions}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{men}\:\mathrm{and}\:\mathrm{women}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{committee}. \\…

Find-the-number-of-ways-the-digits-0-1-2-and-3-can-be-permuted-to-give-rise-to-a-number-greater-than-2000-

Question Number 17770 by chux last updated on 10/Jul/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{the} \\ $$$$\mathrm{digits}\:\mathrm{0},\mathrm{1},\mathrm{2}\:\mathrm{and}\:\mathrm{3}\:\mathrm{can}\:\mathrm{be}\:\mathrm{permuted} \\ $$$$\mathrm{to}\:\mathrm{give}\:\mathrm{rise}\:\mathrm{to}\:\mathrm{a}\:\mathrm{number}\:\mathrm{greater} \\ $$$$\mathrm{than}\:\mathrm{2000}. \\ $$ Answered by alex041103 last updated on 10/Jul/17…