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


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

$C a l l a 7 - d i g i t t e l e p h o n e n u m b e r d_{1} d_{2} d_{3} - d_{4} d_{5} d_{6} d_{7}$ $m e m o r a b l e i f t h e p r e f i x s e q u e n c e d_{1} d_{2} d_{3}$ $i s e x a c t l y t h e s a m e a s e i t h e r o f t h e s e q u e n c e s$ $d_{4} d_{5} d_{6} o r d_{5} d_{6} d_{7} (p o s i b l y b o t h) .$ $A s s u m i n g t h a t e a c h d_{i} c a n b e a n y o f t h e t e n$ $d e c i m a l d i g i t s 0, 1, 2, . . ., 9 . F i n d t h e n u m b e r$ $o f d i f f e r e n t m e m o r a b l e t e l e p h o n e$ $n u m b e r s$
Commented by mr W last updated on 01/Nov/20

$2 \times C_{3}^{10} \times 3! \times 7 = 10080$
Commented by bemath last updated on 01/Nov/20

$thank you sir$
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