Question Number 111394 by Aina Samuel Temidayo last updated on 03/Sep/20
$$\mathrm{A}\:\mathrm{teacher}\:\mathrm{conducts}\:\mathrm{a}\:\mathrm{test}\:\mathrm{for}\:\mathrm{five} \\ $$$$\mathrm{students}.\:\mathrm{He}\:\mathrm{provides}\:\mathrm{the}\:\mathrm{marking} \\ $$$$\mathrm{scheme}\:\mathrm{and}\:\mathrm{asked}\:\mathrm{them}\:\mathrm{to}\:\mathrm{exchange} \\ $$$$\mathrm{their}\:\mathrm{scripts}\:\mathrm{such}\:\mathrm{that}\:\mathrm{none}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{marks}\:\mathrm{his}\:\mathrm{own}\:\mathrm{script}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{ways} \\ $$$$\mathrm{can}\:\mathrm{the}\:\mathrm{students}\:\mathrm{carry}\:\mathrm{out}\:\mathrm{the} \\ $$$$\mathrm{marking}? \\ $$
Commented by mr W last updated on 03/Sep/20
$$\mathrm{44} \\ $$
Commented by Aina Samuel Temidayo last updated on 03/Sep/20
$$\mathrm{Solution}\:\mathrm{please}? \\ $$
Answered by mr W last updated on 03/Sep/20
$${total}:\:\mathrm{5}! \\ $$$${at}\:{least}\:{one}\:{student}\:{marks}\:{his}\:{own}\:{script}: \\ $$$${C}_{\mathrm{1}} ^{\mathrm{5}} ×\mathrm{4}! \\ $$$${at}\:{least}\:{two}\:{students}\:{mark}\:{their}\:{own}\:{scripts}: \\ $$$${C}_{\mathrm{2}} ^{\mathrm{5}} ×\mathrm{3}! \\ $$$$… \\ $$$$\Rightarrow\mathrm{5}!−{C}_{\mathrm{1}} ^{\mathrm{5}} ×\mathrm{4}!+{C}_{\mathrm{2}} ^{\mathrm{5}} ×\mathrm{3}!−{C}_{\mathrm{3}} ^{\mathrm{5}} ×\mathrm{2}!+{C}_{\mathrm{4}} ^{\mathrm{5}} ×\mathrm{1}!−\mathrm{1} \\ $$$$=\mathrm{120}−\mathrm{120}+\mathrm{60}−\mathrm{20}+\mathrm{5}−\mathrm{1} \\ $$$$=\mathrm{44} \\ $$
Commented by Aina Samuel Temidayo last updated on 03/Sep/20
$$\mathrm{General}\:\mathrm{formula}\:\mathrm{for}\:!\mathrm{n}\:? \\ $$
Commented by mr W last updated on 03/Sep/20
$${or} \\ $$$$!\mathrm{5}=\mathrm{44} \\ $$
Commented by mr W last updated on 03/Sep/20
Commented by mr W last updated on 03/Sep/20
$${as}\:{shown}\:{above} \\ $$$$!{n}={n}!\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}!} \\ $$$${we}\:{have} \\ $$$$!{n}=\left({n}−\mathrm{1}\right)\left[!\left({n}−\mathrm{1}\right)+!\left({n}−\mathrm{2}\right)\right] \\ $$$$!{n}=\left[\frac{{n}!}{{e}}\right] \\ $$
Commented by mr W last updated on 03/Sep/20
$${we}\:{see} \\ $$$$\frac{!{n}}{{n}!}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}!} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{!{n}}{{n}!}=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}!}=\frac{\mathrm{1}}{{e}} \\ $$
Commented by Aina Samuel Temidayo last updated on 04/Sep/20
$$\mathrm{Thanks}. \\ $$