Category: Permutation and Combination

Question Number 124709 by benjo_mathlover last updated on 05/Dec/20 $${Between}\:\mathrm{20000}\:{and}\:\mathrm{70000}\: \\ $$$${find}\:{the}\:{number}\:{of}\:{even}\:{integers} \\ $$$${in}\:{which}\:{no}\:{digits}\:{is}\:{repeated} \\ $$ Answered by liberty last updated on 05/Dec/20 $${Let}\:{abcde}\:{be}\:{a}\:{required}\:{even}\:{integer}.\: \\…

Question Number 59021 by mr W last updated on 03/May/19 $$\left[{Q}\mathrm{58885}\:{reposted}\right] \\ $$$${How}\:{many}\:\mathrm{4}−{digit}\:{numbers}\:{abcd}\:{exist} \\ $$$${which}\:{are}\:{divisible}\:{by}\:\mathrm{3}\:{and}\:{satisfy} \\ $$$${a}\leqslant{b}\leqslant{c}\leqslant{d}? \\ $$ Commented by tanmay last updated on…

Question Number 124372 by mr W last updated on 02/Dec/20 $${How}\:{many}\:{six}-{digit}\:{numbers}\:{contain} \\ $$$${exactly}\:{six}\:{different}\:{digits}? \\ $$ Answered by bemath last updated on 02/Dec/20 $$=\:\mathrm{9}×{P}_{\mathrm{5}} ^{\:\mathrm{9}} \:=\:\mathrm{9}×\frac{\mathrm{9}!}{\mathrm{4}!}…

Question Number 124173 by bramlexs22 last updated on 01/Dec/20 Answered by bobhans last updated on 01/Dec/20 $$\left(\mathrm{1}+{kx}\right)\left(\mathrm{1}−\mathrm{2}{x}\right)^{\mathrm{5}} \:=\:\left(\mathrm{1}+{kx}\right)\left[\underset{{n}=\mathrm{0}} {\overset{\mathrm{5}} {\sum}}{C}_{{n}} ^{\:\mathrm{5}} \left(−\mathrm{2}{x}\right)^{\mathrm{5}−{n}} \:\right] \\ $$$$=\left(\mathrm{1}+{kx}\right)\left({C}_{\mathrm{3}}…

Question Number 58373 by Tawa1 last updated on 22/Apr/19 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\:\mathrm{x}^{\mathrm{6}} \:\:\mathrm{in}\:\:\:\left(\mathrm{2x}\:+\:\mathrm{1}\right)^{\mathrm{6}} \:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} \\ $$ Commented by maxmathsup by imad last updated on 24/Apr/19 $${we}\:{have}\:\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{6}}…