# 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

363.3k+ views

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.

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.

Last updated date: 29th Sep 2023

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