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Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either $635$ or $674,$ the number is odd, and the number nine appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?

  1. $1000$
  2. $2430$
  3. $3402$
  4. $3006$
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      635_ _ _ _

OR

      674_ _ _ _

First 3 digit 4th digit 5th digit 6th digit 7th digit No. of trial & error
635 1 way (can place only no. 9) 9 ways (can place 0 to 8 no.s) 9 ways (can place 0 to 8 no.s) 4 ways (can place 1/3/5/7) 1*9*9*4 = 324
635 9 ways (can place 0 to 8 no.s) 1 way (can place only no. 9) 9 ways (can place 0 to 8 no.s) 4 ways (can place 1/3/5/7) 9*1*9*4 = 324
635 9 ways (can place 0 to 8 no.s) 9 ways (can place 0 to 8 no.s) 1 way (can place only no. 9) 4 ways (can place 1/3/5/7) 9*9*1*4 = 324
635 9 ways (can place 0 to 8 no.s) 9 ways (can place 0 to 8 no.s) 9 ways (can place 0 to 8 no.s) 1 way (can place only no. 9) 9*9*9*1 = 729

The total no. of Trial and Error process with 635 as prefix is (324 + 324 + 324 + 729) = 1701.

This 1701 combinations will repeat with 674 as prefix also.

Minimum no. of trials = 1701 + 1701

                                  = 3402 (option 3)

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