Question Number 56743 by Tawa1 last updated on 22/Mar/19
Answered by mr W last updated on 23/Mar/19
$$\left(\mathrm{14}\right) \\ $$$${all}\:{possibilities}:\:{C}_{\mathrm{5}} ^{\mathrm{50}} \\ $$$${without}\:{teacher}:\:{C}_{\mathrm{5}} ^{\mathrm{45}} \\ $$$$\Rightarrow{C}_{\mathrm{5}} ^{\mathrm{50}} −{C}_{\mathrm{5}} ^{\mathrm{45}} =\mathrm{897001} \\ $$$$ \\ $$$$\left(\mathrm{15}\right) \\ $$$${C}_{\mathrm{2}} ^{\mathrm{15}} ×\mathrm{7}!=\mathrm{529200} \\ $$
Commented by mr W last updated on 23/Mar/19
$$\left(\mathrm{14}\right) \\ $$$${alternative}: \\ $$$$\underset{\mathrm{1}\:{teacher}} {{C}_{\mathrm{1}} ^{\mathrm{5}} ×{C}_{\mathrm{4}} ^{\mathrm{45}} }+\underset{\mathrm{2}\:{teachers}} {{C}_{\mathrm{2}} ^{\mathrm{5}} ×{C}_{\mathrm{3}} ^{\mathrm{45}} }+\underset{\mathrm{3}\:{teachers}} {{C}_{\mathrm{3}} ^{\mathrm{5}} ×{C}_{\mathrm{2}} ^{\mathrm{45}} }+\underset{\mathrm{4}\:{teachers}} {{C}_{\mathrm{4}} ^{\mathrm{5}} ×{C}_{\mathrm{1}} ^{\mathrm{45}} }+\underset{\mathrm{5}\:{teachers}} {{C}_{\mathrm{5}} ^{\mathrm{5}} } \\ $$$$=\mathrm{744975}+\mathrm{141900}+\mathrm{9900}+\mathrm{225}+\mathrm{1} \\ $$$$=\mathrm{897001} \\ $$