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Question Number 206156 by cortano21 last updated on 08/Apr/24
Commented by mr W last updated on 08/Apr/24
$${what}\:{i}\:{think}\:{is}\:{clear}: \\ $$$${in}\:{the}\:{package}\:{there}\:{must}\:{be}\:{exactly} \\ $$$${one}\:{egg}.\:{there}\:{must}\:{be}\:{at}\:{least}\:{three} \\ $$$${lettuces}.\:{there}\:{must}\:{be}\:{at}\:{least}\:{one} \\ $$$${tofu},\:{at}\:{least}\:{one}\:{potato}.\:{bitter}\:{melon} \\ $$$${may}\:{be}\:{omitted}.\:{cabbage}\:{may}\:{be} \\ $$$${omitted}. \\ $$$${what}\:{is}\:{unclear}: \\ $$$${the}\:{maximum}\:{number}\:{of}\:{bitter}\:{melon} \\ $$$${and}\:{cabbage}\:{is}\:\mathrm{1}.\:{does}\:{it}\:{mean}: \\ $$$${totally}\:{there}\:{is}\:{at}\:{most}\:{one}\:{bitter} \\ $$$${melon}\:{or}\:{one}\:{cabbage}? \\ $$$${i}.{e}.\:{bitter}\:{melon}+{cabbage}\leqslant\mathrm{1}. \\ $$$${or} \\ $$$${there}\:{is}\:{at}\:{most}\:{one}\:{bitter}\:{melon} \\ $$$${and}\:{at}\:{most}\:{one}\:{cabbage}? \\ $$$${i}.{e}.\:{bitter}\:{melon}\leqslant\mathrm{1}\:{and}\:{cabbage}\leqslant\mathrm{1}. \\ $$
Commented by cortano21 last updated on 09/Apr/24
$${yes}\:{sir} \\ $$
Commented by mr W last updated on 09/Apr/24
$${i}\:{asked}\:{what}\:{is}\:{meant},\:{option}\:\mathrm{1}\:{or} \\ $$$${option}\:\mathrm{2}.\:{you}\:{answered}\:{yes}.\:{this}\:{is} \\ $$$${not}\:{a}\:{proper}\:{answer}. \\ $$
Commented by mr W last updated on 09/Apr/24
$${let}'{s}\:{assume}\:{that}\:{option}\:\mathrm{1}\:{is}\:{meant}, \\ $$$${i}.{e}.\:{bitter}\:{melon}+{cabbage}\leqslant\mathrm{1}. \\ $$
Answered by mr W last updated on 09/Apr/24
$${L}={number}\:{of}\:{lettuces},\:{L}\geqslant\mathrm{3} \\ $$$${T}={number}\:{of}\:{tofu},\:{T}\geqslant\mathrm{1} \\ $$$${E}={number}\:{of}\:{eggs},\:{E}=\mathrm{1} \\ $$$${P}={number}\:{of}\:{potatoes},\:{P}\geqslant\mathrm{1} \\ $$$${C}={number}\:{of}\:{cabbage},\:\mathrm{0}\leqslant{C} \\ $$$${B}={number}\:{of}\:{bitter}\:{lemons},\:\mathrm{0}\leqslant{B} \\ $$$$\:\:\:\:\:\:\:\:{but}\:{K}={B}+{C}\leqslant\mathrm{1} \\ $$$${L}+{T}+{E}+{P}+{C}+{B}=\mathrm{10} \\ $$$${L}+{T}+{P}+{K}=\mathrm{9} \\ $$$$\underset{{L}} {\left({x}^{\mathrm{3}} +{x}^{\mathrm{4}} +...\right)}\underset{{T},\:{P}} {\left({x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\right)^{\mathrm{2}} }\underset{{K}={B}+{C}} {\left(\mathrm{1}+\mathrm{2}{x}\right)} \\ $$$$=\left(\mathrm{1}+\mathrm{2}{x}\right){x}^{\mathrm{5}} \left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\right)^{\mathrm{3}} \\ $$$$=\frac{\left(\mathrm{1}+\mathrm{2}{x}\right){x}^{\mathrm{5}} }{\left(\mathrm{1}−{x}\right)^{\mathrm{3}} }=\left(\mathrm{1}+\mathrm{2}{x}\right){x}^{\mathrm{5}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} {x}^{{k}} \\ $$$${for}\:{x}^{\mathrm{9}} \:{term}: \\ $$$${k}=\mathrm{4}:\:{C}_{\mathrm{2}} ^{\mathrm{6}} =\mathrm{15} \\ $$$${k}=\mathrm{3}:\:\mathrm{2}{C}_{\mathrm{2}} ^{\mathrm{5}} =\mathrm{20} \\ $$$$\Rightarrow\:{coef}.\:{of}\:{x}^{\mathrm{9}} \:{is}\:\mathrm{15}+\mathrm{20}=\mathrm{35} \\ $$$${that}\:{means}\:{there}\:{are}\:\mathrm{35}\:{ways}\:{to} \\ $$$${choose}\:{the}\:{content}\:{of}\:{food}\:{package}. \\ $$