Question Number 222889 by Tawa11 last updated on 10/Jul/25
Commented by mr W last updated on 10/Jul/25
$${such}\:{a}\:{bench}: \\ $$
Commented by mr W last updated on 10/Jul/25
Commented by mr W last updated on 10/Jul/25
$${or}\:{such}\:{a}\:{bench}: \\ $$
Commented by mr W last updated on 10/Jul/25
Commented by Tawa11 last updated on 10/Jul/25
$$\mathrm{Does}\:\mathrm{the}\:\mathrm{bench}\:\mathrm{type}\:\mathrm{matters}\:\mathrm{sir}.\:\mathrm{Hahahahaha}! \\ $$
Commented by mr W last updated on 10/Jul/25
$${what}\:{do}\:{you}\:{think}?\:{is}\:{this}\:{couple} \\ $$$${seated}\:{in}\:\mathrm{4}\:{different}\:{ways},\:{or}\:{just} \\ $$$${in}\:{one}\:{same}\:{way}? \\ $$
Commented by mr W last updated on 10/Jul/25
Commented by Tawa11 last updated on 10/Jul/25
$$\mathrm{I}\:\mathrm{never}\:\mathrm{thought}\:\mathrm{of}\:\mathrm{these}\:\mathrm{sittings}. \\ $$
Commented by Tawa11 last updated on 10/Jul/25
$$\mathrm{I}\:\mathrm{will}\:\mathrm{love}\:\mathrm{to}\:\mathrm{know}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\mathrm{each}\:\mathrm{bench}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{the}\:\mathrm{second}\:\mathrm{bench}\:\mathrm{has} \\ $$$$\mathrm{dufferent}\:\mathrm{cases}. \\ $$$$ \\ $$$$\mathrm{But}\:\mathrm{the}\:\mathrm{question}\:\mathrm{is}\:\mathrm{the}\:\mathrm{bench}\:\mathrm{with} \\ $$$$\mathrm{back}.\:\:\mathrm{First}\:\mathrm{bench} \\ $$
Commented by fantastic last updated on 10/Jul/25
$${WOW}!\: \\ $$$${Just}\:{nice}! \\ $$
Answered by mr W last updated on 10/Jul/25
$$\underline{{bench}\:{with}\:{back}} \\ $$$$\left({a}\right) \\ $$$$\mathrm{7}!×\mathrm{2}=\mathrm{10080}\:{ways} \\ $$$$ \\ $$$$\left({b}\right) \\ $$$$\mathrm{6}!×{C}_{\mathrm{2}} ^{\mathrm{7}} ×\mathrm{2}=\mathrm{30240}\:{ways} \\ $$$${or} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)^{\mathrm{2}} \left({x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +…\right) \\ $$$$=\frac{{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{3}} }={x}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} {x}^{{k}} \\ $$$${coef}.\:{of}\:{x}^{\mathrm{6}} \:{is}\:{C}_{\mathrm{2}} ^{\mathrm{5}+\mathrm{2}} ,\:{so}\:{the}\:{answer}\:{is} \\ $$$$\mathrm{2}×{C}_{\mathrm{2}} ^{\mathrm{5}+\mathrm{2}} ×\mathrm{6}!=\mathrm{30240} \\ $$
Commented by mr W last updated on 10/Jul/25
$$\underline{{bench}\:{without}\:{back}} \\ $$$$=\:{bench}\:{with}\:{back}\:×\mathrm{2}^{\mathrm{8}} \\ $$
Commented by Tawa11 last updated on 10/Jul/25
$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{appreciate}. \\ $$