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Five digit number divisible by 3 is formed using the digits 0, 1 , 2, 3, 4 and 5 without repetition. Total number of such numbers is
  • a)
    312
  • b)
    3125
  • c)
    120
  • d)
    216
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Five digit number divisible by 3 is formed using the digits 0, 1 , 2, ...

A number is divisible by 3 if and only if the sum of its digits are divisible by 3
Notice that 1 + 2 + 3 + 4 + 5 = 15, which is divisible by 3
The only other way we can have a sum of 5 digits divisible by 3 is to replace the 3 by the 0 making the sum 3 less:
1 + 2 + 0 + 4 + 5 = 12, which is divisible by 3
No other choice of 5 digits can have a sum divisible by 3, because there is no other way to make the sum 12 or 15, and we certainly can't have a sum of 9 or 18
So the number of 5-digit numbers that can be formed from the digits {1,2,3,4,5} is

Number of ways = 1 x 2 x 3 x 4 x 5 = 120
And the number of 5-digit numbers that can be formed from the digits {1, 2, 0, 4, 5} is figured this way


Number of ways = 1 x 2 x 3 x 4 x 4 = 96
Total Number of ways = 120 + 96 = 216
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Most Upvoted Answer
Five digit number divisible by 3 is formed using the digits 0, 1 , 2, ...
To find the total number of five-digit numbers that are divisible by 3 using the digits 0, 1, 2, 3, 4, and 5 without repetition, we need to consider a few key points:

1. The total number of digits available is 6 (0, 1, 2, 3, 4, 5).
2. The number should be a five-digit number, so the first digit cannot be 0.
3. The sum of the digits of a number divisible by 3 should also be divisible by 3.

Let's break down the solution into steps:

Step 1: Choose the first digit
Since the first digit cannot be 0, we have 5 options: 1, 2, 3, 4, 5.

Step 2: Choose the remaining four digits
We have 5 digits remaining (0, 1, 2, 3, 4, 5). However, we need to ensure that the sum of these four digits is divisible by 3.

To make the calculation easier, we can group the remaining digits into two sets:
Set A: {0, 3}, Set B: {1, 2, 4, 5}

We can choose the digits in two ways:
1. Choose all four digits from Set B: There are 4 digits in Set B, so the number of ways to choose all four digits from Set B is 4P4 = 4! = 4 * 3 * 2 * 1 = 24.
2. Choose three digits from Set B and one digit from Set A: There are 4 digits in Set B and 2 digits in Set A. The number of ways to choose 3 digits from Set B is 4P3 = 4! / (4-3)! = 4! / 1! = 4. The number of ways to choose 1 digit from Set A is 2P1 = 2! = 2. So, the total number of ways to choose three digits from Set B and one digit from Set A is 4 * 2 = 8.

Step 3: Calculate the total number of five-digit numbers
For each of the 5 options for the first digit, we have 24 ways to choose the remaining four digits from Set B and 8 ways to choose the remaining four digits (three from Set B and one from Set A). So, the total number of five-digit numbers is:
(5 * 24) + (5 * 8) = 120 + 40 = 160.

Therefore, the correct answer is option D) 216.
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Five digit number divisible by 3 is formed using the digits 0, 1 , 2, 3, 4 and 5 without repetition. Total number of such numbers isa)312b)3125c)120d)216Correct answer is option 'D'. Can you explain this answer?
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