Question Number 102243 by mathocean1 last updated on 07/Jul/20
$${At}\:{examination},\:{a}\:{student}\:{most} \\ $$$${answer}\:{to}\:\mathrm{8}\:{question}\:{betwen}\:{a}\:{total} \\ $$$${of}\:\mathrm{10}\:{question}. \\ $$$$\left.\mathrm{1}\right)\:{How}\:{many}\:{many}\:{choices}\:{is}\: \\ $$$${possible}? \\ $$$$\left.\mathrm{2}\right){How}\:{many}\:{choices}\:{is}\:{possible}\:{if}\:{a} \\ $$$${student}\:{most}\:{answer}\:{to}\:\mathrm{3}\:{questions} \\ $$$${at}\:{least}? \\ $$$$\left.\mathrm{3}\right){How}\:{many}\:{choices}\:{is}\:{possible}\:{if}\:{a} \\ $$$${student}\:{most}\:{answer}\:{to}\:{at}\:{least}\:\mathrm{4} \\ $$$${questions}\:{of}\:\:{the}\:{five}\:{first}\:{questions}? \\ $$
Answered by mr W last updated on 07/Jul/20
$$\left.\mathrm{1}\right) \\ $$$${to}\:{select}\:\mathrm{8}\:{from}\:\mathrm{10}\:{questions}\:{there}\:{are} \\ $$$${C}_{\mathrm{8}} ^{\mathrm{10}} =\mathrm{45}\:{possibilities}. \\ $$$$\left.\mathrm{2}\right) \\ $$$${do}\:{you}\:{mean}\:\mathrm{3}\:{certain}\:{questions}\:{must} \\ $$$${be}\:{selected}?\:{then}\:{one}\:{can}\:{select}\:\mathrm{5}\:{from} \\ $$$${the}\:{remaining}\:\mathrm{7}\:{questions},\:{there}\:{are} \\ $$$${C}_{\mathrm{5}} ^{\mathrm{7}} =\mathrm{21}\:{ways}. \\ $$$$\left.\mathrm{3}\right) \\ $$$${to}\:{select}\:{at}\:{least}\:\mathrm{4}\:{from}\:{the}\:{first}\:\mathrm{5} \\ $$$${questions},\:{there}\:{are} \\ $$$${C}_{\mathrm{4}} ^{\mathrm{5}} \:{ways},\:{then}\:{select}\:{another}\:\mathrm{4}\:{from} \\ $$$${the}\:{remaining}\:\mathrm{6}\:{questions},\:{there}\:{are} \\ $$$${C}_{\mathrm{4}} ^{\mathrm{6}} \:{ways},\:{totally} \\ $$$${C}_{\mathrm{4}} ^{\mathrm{5}} ×{C}_{\mathrm{4}} ^{\mathrm{6}} =\mathrm{75}\:{ways}. \\ $$