Question

If a five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4, and 5 without repetition, then the total number of ways this can be done is:

• A. 120
• B. 144
• C. 192
• D. 216

Hint:

When forming numbers divisible by 3, check the sum of the digits and make sure the sum is divisible by 3. For five-digit numbers, be cautious when placing 0 as the first digit.

Answer 1

The Correct Option is D

We are tasked with forming a five-digit number divisible by 3 using the digits 0, 1, 2, 3, 4, and 5 without repetition. A number is divisible by 3 if the sum of its digits is divisible by 3. 
Step 1: Total sum of the digits The sum of the digits 0, 1, 2, 3, 4, and 5 is: \[ 0 + 1 + 2 + 3 + 4 + 5 = 15. \] Since the total sum of the digits is 15, which is divisible by 3, the sum of the digits of any five-digit number formed from these digits will also be divisible by 3, provided that we leave out one of the digits. 
Step 2: Choosing one digit to leave out We can leave out any of the six digits (0, 1, 2, 3, 4, 5). If we leave out a digit, the sum of the remaining digits will still be divisible by 3. We need to calculate the number of five-digit numbers that can be formed with the remaining digits. 
Step 3: Counting the number of five-digit numbers We must choose 5 digits from the 6 available digits. We have 6 choices for the digit to leave out. For each selection of 5 digits, the number of ways to arrange them is given by the number of permutations of 5 digits. However, the first digit cannot be 0, so we need to adjust the counting. - If 0 is not selected, all 5 digits can be arranged in \( 5! \) ways. - If 0 is selected, the first digit cannot be 0, so we must select the first digit from the remaining 4 digits (1, 2, 3, 4, or 5) and arrange the other 4 digits. 
Step 4: Calculating the number of arrangements - If 0 is not selected, we have \( 5! = 120 \) ways. - If 0 is selected, the number of valid arrangements is \( 4 \times 4! = 4 \times 24 = 96 \). Thus, the total number of arrangements is: \[ 120 + 96 = 216. \] Thus, the total number of ways to form a five-digit number divisible by 3 is \( 216 \).