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Question Number 56743 by Tawa1 last updated on 22/Mar/19

Answered by mr W last updated on 23/Mar/19

(14)  all possibilities: C_5 ^(50)   without teacher: C_5 ^(45)   ⇒C_5 ^(50) −C_5 ^(45) =897001    (15)  C_2 ^(15) ×7!=529200

$$\left(\mathrm{14}\right) \\ $$$${all}\:{possibilities}:\:{C}_{\mathrm{5}} ^{\mathrm{50}} \\ $$$${without}\:{teacher}:\:{C}_{\mathrm{5}} ^{\mathrm{45}} \\ $$$$\Rightarrow{C}_{\mathrm{5}} ^{\mathrm{50}} −{C}_{\mathrm{5}} ^{\mathrm{45}} =\mathrm{897001} \\ $$$$ \\ $$$$\left(\mathrm{15}\right) \\ $$$${C}_{\mathrm{2}} ^{\mathrm{15}} ×\mathrm{7}!=\mathrm{529200} \\ $$

Commented by mr W last updated on 23/Mar/19

(14)  alternative:  C_1 ^5 ×C_4 ^(45) _(1 teacher) +C_2 ^5 ×C_3 ^(45) _(2 teachers) +C_3 ^5 ×C_2 ^(45) _(3 teachers) +C_4 ^5 ×C_1 ^(45) _(4 teachers) +C_5 ^5 _(5 teachers)   =744975+141900+9900+225+1  =897001

$$\left(\mathrm{14}\right) \\ $$$${alternative}: \\ $$$$\underset{\mathrm{1}\:{teacher}} {{C}_{\mathrm{1}} ^{\mathrm{5}} ×{C}_{\mathrm{4}} ^{\mathrm{45}} }+\underset{\mathrm{2}\:{teachers}} {{C}_{\mathrm{2}} ^{\mathrm{5}} ×{C}_{\mathrm{3}} ^{\mathrm{45}} }+\underset{\mathrm{3}\:{teachers}} {{C}_{\mathrm{3}} ^{\mathrm{5}} ×{C}_{\mathrm{2}} ^{\mathrm{45}} }+\underset{\mathrm{4}\:{teachers}} {{C}_{\mathrm{4}} ^{\mathrm{5}} ×{C}_{\mathrm{1}} ^{\mathrm{45}} }+\underset{\mathrm{5}\:{teachers}} {{C}_{\mathrm{5}} ^{\mathrm{5}} } \\ $$$$=\mathrm{744975}+\mathrm{141900}+\mathrm{9900}+\mathrm{225}+\mathrm{1} \\ $$$$=\mathrm{897001} \\ $$

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