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Question Number 120204 by bemath last updated on 30/Oct/20

Call a 7−digit telephone number d_1 d_2 d_3 −d_4 d_5 d_6 d_7   memorable if the prefix sequence d_1 d_2 d_3   is exactly the same as either of the sequences  d_4 d_5 d_6  or d_5 d_6 d_7  (posibly both).  Assuming that each d_i  can be any of the ten  decimal digits 0,1,2,...,9 . Find the number  of different memorable telephone  numbers

$${Call}\:{a}\:\mathrm{7}−{digit}\:{telephone}\:{number}\:{d}_{\mathrm{1}} {d}_{\mathrm{2}} {d}_{\mathrm{3}} −{d}_{\mathrm{4}} {d}_{\mathrm{5}} {d}_{\mathrm{6}} {d}_{\mathrm{7}} \\ $$$${memorable}\:{if}\:{the}\:{prefix}\:{sequence}\:{d}_{\mathrm{1}} {d}_{\mathrm{2}} {d}_{\mathrm{3}} \\ $$$${is}\:{exactly}\:{the}\:{same}\:{as}\:{either}\:{of}\:{the}\:{sequences} \\ $$$${d}_{\mathrm{4}} {d}_{\mathrm{5}} {d}_{\mathrm{6}} \:{or}\:{d}_{\mathrm{5}} {d}_{\mathrm{6}} {d}_{\mathrm{7}} \:\left({posibly}\:{both}\right). \\ $$$${Assuming}\:{that}\:{each}\:{d}_{{i}} \:{can}\:{be}\:{any}\:{of}\:{the}\:{ten} \\ $$$${decimal}\:{digits}\:\mathrm{0},\mathrm{1},\mathrm{2},...,\mathrm{9}\:.\:{Find}\:{the}\:{number} \\ $$$${of}\:{different}\:{memorable}\:{telephone} \\ $$$${numbers} \\ $$

Commented by mr W last updated on 01/Nov/20

2×C_3 ^(10) ×3!×7=10080

$$\mathrm{2}×{C}_{\mathrm{3}} ^{\mathrm{10}} ×\mathrm{3}!×\mathrm{7}=\mathrm{10080} \\ $$

Commented by bemath last updated on 01/Nov/20

thank you sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$

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