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Question: A five digit number divisible by 3 is to be formed using the digits 0,1,2,3,4 and 5, without repetition. The total number of ways this can be done, is
A. 216
B. 240
C. 600
D. 3125

Answer Verified Verified
Hint: A number to be divisible by 3, the sum of all the digits should be divisible by 3.
In this question, we are supposed to form a five digit number which will be divisible by 3, and the divisibility test of 3 is the sum of digits should be divisible by 3.
Therefore, we are only going to consider 5 digits whose sum will result in a number which will be divisible by 3.

Complete step-by-step answer:

Let us observe the digits given to us, we have 0,1,2,3,4 and 5, to make a five digit number we only need 5 digits out of the given 6 digits,
Case 1: Using digits 0,1,2,4 and 5.
The number of ways in which we can arrange these 5 digits are \[ \Rightarrow 4 \times 4 \times 3 \times 2 \times 1\]
                                                \[ \Rightarrow 96\]

Case 2: Using the digits 1,2,3,4 and 5
The number of ways in which we can arrange these 5 digits are \[ \Rightarrow 5 \times 4 \times 3 \times 2 \times 1\]
                                                      \[ \Rightarrow 120\]
Therefore, the total number of cases = 96+120
                                                                   =216

Therefore, Option A is the correct answer.

Note: Make sure that you do not take 0 in the first place because that will make the number a 4-digit number which will be considered wrong as we are supposed to form a 5-digit number.
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