Previous in Permutation and Combination Next in Permutation and Combination
Question Number 119747 by benjo_mathlover last updated on 26/Oct/20
$${Suppose}\:{that}\:\mathrm{7}\:{blue}\:{balls}\:,\:\mathrm{8}\:{red}\:{balls}\:{and}\:\mathrm{9}\:{green} \\ $$$${balls}\:{should}\:{be}\:{put}\:{into}\:{three}\:{boxes}\:{labeled} \\ $$$$\mathrm{1},\mathrm{2}\:{and}\:\mathrm{3},\:{so}\:{that}\:{any}\:{box}\:{contains}\:{at}\:{least} \\ $$$${one}\:{balls}\:{of}\:{each}\:{colour}.\:{How}\:{many}\:{ways} \\ $$$${can}\:{this}\:{arrangement}\:{be}\:{done}? \\ $$
Answered by mr W last updated on 26/Oct/20
$${to}\:{divide}\:\mathrm{7}\:{blue}\:{balls}\:{into}\:\mathrm{3}\:{groups}, \\ $$$${using}\:{stars}\:\&\:{bars}\:{method}, \\ $$$$\square\mid\square\mid\square\mid\square\mid\square\mid\square\mid\square \\ $$$${there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{6}} \:{ways}. \\ $$$${similarly}\:{to}\:{divide}\:\mathrm{8}\:{red}\:{balls}\:{into} \\ $$$$\mathrm{3}\:{groups}\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{7}} \:{ways},\:{and}\:{to} \\ $$$${divide}\:\mathrm{9}\:{green}\:{balls}\:{into}\:\mathrm{3}\:{groups} \\ $$$${there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{8}} \:{ways},\:{so}\:{totally}\:{we}\:{have} \\ $$$${C}_{\mathrm{2}} ^{\mathrm{6}} ×{C}_{\mathrm{2}} ^{\mathrm{7}} ×{C}_{\mathrm{2}} ^{\mathrm{8}} =\mathrm{8820}\:{ways}. \\ $$