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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?

- $1000$
- $2430$
- $3402$
- $3006$

1 vote

Best answer

**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**)