Using the digits 0,1,2,3and4,four digit number are made without repetition. What is the probability that the number formed is even ?

To determine the probability that a four-digit number formed using the digits 0, 1, 2, 3, and 4 (without repetition) is even, we first need to understand the conditions for a number to be classified as even. A number is even if its last digit is one of the even digits available in our set. In this case, the even digits are 0, 2, and 4. Let's break this down step by step.

Step 1: Total Possible Four-Digit Numbers

When forming a four-digit number, the first digit cannot be 0 (as it would then be a three-digit number). Therefore, we have the following options for the first digit:

• 1
• 2
• 3
• 4

This gives us 4 choices for the first digit. After choosing the first digit, we have 4 remaining digits to choose from for the second digit, 3 for the third, and 2 for the fourth. Thus, the total number of four-digit combinations can be calculated as follows:

Total combinations = 4 (choices for the first digit) × 4 (choices for the second) × 3 (choices for the third) × 2 (choices for the fourth)

Calculating this gives:

Total combinations = 4 × 4 × 3 × 2 = 96

Step 2: Counting Even Four-Digit Numbers

Next, we need to find out how many of these combinations result in an even number. As mentioned earlier, the last digit must be one of the even digits: 0, 2, or 4. We will analyze each case based on the last digit.

Case 1: Last Digit is 0

If the last digit is 0, the first digit can be 1, 2, 3, or 4 (4 choices). After selecting the first digit, we have 3 remaining digits to choose from for the second digit and 2 for the third. Thus, the number of combinations in this case is:

4 (choices for the first) × 3 (choices for the second) × 2 (choices for the third) = 24

Case 2: Last Digit is 2

If the last digit is 2, the first digit can be 1, 3, or 4 (3 choices, since 0 cannot be the first digit). After selecting the first digit, we have 3 remaining digits (including 0) to choose from for the second digit and 2 for the third. Thus, the number of combinations here is:

3 (choices for the first) × 3 (choices for the second) × 2 (choices for the third) = 18

Case 3: Last Digit is 4

If the last digit is 4, the first digit can be 1, 2, or 3 (3 choices). Similar to the previous case, we have 3 remaining digits (including 0) to choose from for the second digit and 2 for the third. Thus, the number of combinations in this case is:

3 (choices for the first) × 3 (choices for the second) × 2 (choices for the third) = 18

Step 3: Total Even Combinations

Now, we can sum the combinations from all three cases to find the total number of even four-digit numbers:

Total even combinations = 24 (last digit 0) + 18 (last digit 2) + 18 (last digit 4) = 60

Step 4: Calculating the Probability

The probability of forming an even four-digit number is the ratio of the number of even combinations to the total combinations:

Probability = Total even combinations / Total combinations = 60 / 96

To simplify this fraction, we can divide both the numerator and the denominator by 12:

Probability = 5 / 8

Final Result

The probability that a four-digit number formed using the digits 0, 1, 2, 3, and 4 without repetition is even is 5/8.