Question

Sameer has to make a telephone call to his friend Harish, unfortunately he does not remember the 7-digit phone number. But he remembers that the first three digits are 635 or 674 the number is odd and there is exactly one 9 in the number. Find the maximum number of trials that Sameer has to make to be successful.

Answer 1

In the given question for the first three digits combination, we have two choices one is 635 or 674. Also, we are provided with the information that the phone numbers contain the digit 9 and it occurs only once.

Complete step-by-step solution:
In the given question we need to fill the 7-digit phone number from 0-9. We also know that the number is odd therefore we can have 1,3,5,7,9 at the unit places. Also, out of the seven digits we are already given the three digits so we are left to find the four digits from seven digits number.
Now, if we put one 9 in a unit place then we remain with three more digits to be filled and the choices we can make are possible in $9\times 9\times 9\times 1=729$ ways as we can use digits from 0-8 to fill the remaining three digits.
Secondly, we can choose a unit place from 1,3,5,7 and also, we need to include 9 which can be placed in three places and rest both places can be filled with any digit from 0-8. Therefore, all possible ways are $4\times 3\times 9\times 9=972$
So, the total number of ways in which the 7-digit number can be written is 972+729 which is 1701 when we choose the first three digits as 635 and similarly another 1701 ways if we choose the first three digits as 674.
Therefore, the total number of ways in which numbers can be formed is 3402..

Note: In the given question we have used the sum rule to answer the possible ways we can also use the product rule that involves permutation of non repeated numbers or positions. Also, combinations can be used to reduce calculations.