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Question-224399

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Question Number 224399 by Mingma last updated on 08/Sep/25
Answered by mr W last updated on 08/Sep/25
(a) C_5 ^(10) ×5!=30240 (b) C_5 ^8 ×5!+C_4 ^8 ×5!+C_3 ^8 ×((5!)/(2!))=18480
$$\left({a}\right) \\ $$$${C}_{\mathrm{5}} ^{\mathrm{10}} ×\mathrm{5}!=\mathrm{30240} \\ $$$$\left({b}\right) \\ $$$${C}_{\mathrm{5}} ^{\mathrm{8}} ×\mathrm{5}!+{C}_{\mathrm{4}} ^{\mathrm{8}} ×\mathrm{5}!+{C}_{\mathrm{3}} ^{\mathrm{8}} ×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{18480} \\ $$
Commented by Mingma last updated on 09/Sep/25
Sir, can you please provide a detailed explanation of how you solved this?
Commented by mr W last updated on 09/Sep/25
(a) to select 5 from 10 pictures there are C_5 ^(10) ways. to arrange these 5 pictures there are 5! ways. so totally there are C_5 ^(10) ×5! ways. (b) case 1: none of the two special pictures is selected, i.e. 5 pictures are selected from the other 8 pictures. there are C_5 ^8 ways to do this. to arrange the 5 pictures there are 5! ways. totally C_5 ^8 ×5!. case 2: one of the two special pictures is selected, i.e. 4 pictures are selected from the other 8 pictures. there are C_4 ^8 ways to do this. to arrange the 5 pictures there are 5! ways. totally C_4 ^8 ×5!. case 3: all of the two special pictures are selected, i.e. 3 pictures are selected from the other 8 pictures. there are C_3 ^8 ways to do this. to arrange the 5 pictures there are ((5!)/(2!)) ways. totally C_3 ^8 ×((5!)/(2!)). all together there are C_5 ^8 ×5!+C_4 ^8 ×5!+C_3 ^8 ×((5!)/(2!))=18480 ways
$$\left({a}\right) \\ $$$${to}\:{select}\:\mathrm{5}\:{from}\:\mathrm{10}\:{pictures}\:{there} \\ $$$${are}\:{C}_{\mathrm{5}} ^{\mathrm{10}} \:{ways}.\:{to}\:{arrange}\:{these}\:\mathrm{5} \\ $$$${pictures}\:{there}\:{are}\:\mathrm{5}!\:{ways}.\:{so}\:{totally} \\ $$$${there}\:{are}\:{C}_{\mathrm{5}} ^{\mathrm{10}} ×\mathrm{5}!\:{ways}. \\ $$$$\left({b}\right) \\ $$$$\underline{{case}\:\mathrm{1}}:\:{none}\:{of}\:{the}\:{two}\:{special}\:{pictures} \\ $$$${is}\:{selected},\:{i}.{e}.\:\mathrm{5}\:{pictures}\:{are}\:{selected} \\ $$$${from}\:{the}\:{other}\:\mathrm{8}\:{pictures}.\:{there}\:{are} \\ $$$${C}_{\mathrm{5}} ^{\mathrm{8}} \:{ways}\:{to}\:{do}\:{this}.\:{to}\:{arrange}\:{the} \\ $$$$\mathrm{5}\:{pictures}\:{there}\:{are}\:\mathrm{5}!\:{ways}.\:{totally} \\ $$$${C}_{\mathrm{5}} ^{\mathrm{8}} ×\mathrm{5}!. \\ $$$$\underline{{case}\:\mathrm{2}}:\:{one}\:{of}\:{the}\:{two}\:{special}\:{pictures} \\ $$$${is}\:{selected},\:{i}.{e}.\:\mathrm{4}\:{pictures}\:{are}\:{selected} \\ $$$${from}\:{the}\:{other}\:\mathrm{8}\:{pictures}.\:{there}\:{are} \\ $$$${C}_{\mathrm{4}} ^{\mathrm{8}} \:{ways}\:{to}\:{do}\:{this}.\:{to}\:{arrange}\:{the} \\ $$$$\mathrm{5}\:{pictures}\:{there}\:{are}\:\mathrm{5}!\:{ways}.\:{totally} \\ $$$${C}_{\mathrm{4}} ^{\mathrm{8}} ×\mathrm{5}!. \\ $$$$\underline{{case}\:\mathrm{3}}:\:{all}\:{of}\:{the}\:{two}\:{special}\:{pictures} \\ $$$${are}\:{selected},\:{i}.{e}.\:\mathrm{3}\:{pictures}\:{are}\:{selected} \\ $$$${from}\:{the}\:{other}\:\mathrm{8}\:{pictures}.\:{there}\:{are} \\ $$$${C}_{\mathrm{3}} ^{\mathrm{8}} \:{ways}\:{to}\:{do}\:{this}.\:{to}\:{arrange}\:{the} \\ $$$$\mathrm{5}\:{pictures}\:{there}\:{are}\:\frac{\mathrm{5}!}{\mathrm{2}!}\:{ways}.\:{totally} \\ $$$${C}_{\mathrm{3}} ^{\mathrm{8}} ×\frac{\mathrm{5}!}{\mathrm{2}!}. \\ $$$${all}\:{together}\:{there}\:{are} \\ $$$${C}_{\mathrm{5}} ^{\mathrm{8}} ×\mathrm{5}!+{C}_{\mathrm{4}} ^{\mathrm{8}} ×\mathrm{5}!+{C}_{\mathrm{3}} ^{\mathrm{8}} ×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{18480}\:{ways} \\ $$
Commented by Mingma last updated on 09/Sep/25
Perfect ��

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