A Dubai based gangster Chhota Vakil is in Switzerland. He wants to rob a bank there, whose locker code according to his information is an odd number between 50 and 450. He also knows that the numbers are from the set: 0, 1, 2, 3, 4, 5}. How many maximum trials he has to take to unlock the locker?Choices:- 54 72 78 106
Question
A Dubai based gangster Chhota Vakil is in Switzerland. He wants to rob a bank there, whose locker code according to his information is an odd number between 50 and 450. He also knows that the numbers are from the set: 0, 1, 2, 3, 4, 5}. How many maximum trials he has to take to unlock the locker?Choices:- 54 72 78 106
Solution 1
The problem is asking for the maximum number of odd numbers that can be formed between 50 and 450 using the digits 0, 1, 2, 3, 4, 5.
Step 1: Identify the possible number of digits for the code. The code is a number between 50 and 450, so it can be either a two-digit number or a three-digit number.
Step 2: Calculate the number of two-digit odd numbers that can be formed. For a two-digit number to be odd, the units place must be an odd digit. From the given set, the odd digits are 1, 3, 5. So, there are 3 possibilities for the units place. For the tens place, any of the 6 digits can be used. So, there are 6 possibilities for the tens place. Therefore, the total number of two-digit odd numbers that can be formed is 3*6 = 18.
Step 3: Calculate the number of three-digit odd numbers that can be formed. For a three-digit number to be odd, again, the units place must be an odd digit. So, there are 3 possibilities for the units place. For the tens place, any of the 6 digits can be used. So, there are 6 possibilities for the tens place. For the hundreds place, 0 cannot be used because the number would not be a three-digit number. So, there are 5 possibilities for the hundreds place. Therefore, the total number of three-digit odd numbers that can be formed is 365 = 90.
Step 4: Add the number of two-digit and three-digit odd numbers to get the total number of trials. 18 + 90 = 108.
However, the numbers 101, 103, 105, 201, 203, 205, 301, 303, 305, 401, 403, 405 are not between 50 and 450. So, we subtract these 12 numbers from the total.
108 - 12 = 96.
But the options provided do not include 96. This suggests that there may be a mistake in the problem or the provided options.
Solution 2
The problem can be solved by considering the possible combinations of the digits given. The locker code is a three-digit odd number between 50 and 450. This means the first digit can be any number from 1 to 4, the second digit can be any number from 0 to 5, and the third digit, since the number has to be odd, can be 1, 3, or 5.
So, the total number of possible combinations is the product of the number of choices for each digit, which is 4 choices for the first digit, 6 choices for the second digit, and 3 choices for the third digit.
Therefore, the maximum number of trials Chhota Vakil has to take to unlock the locker is 463 = 72.
So, the correct answer is 72.
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