How many three digit number can be formed the digits 1,2,3,and 4 if
(1)repetition is allowed
(2) repetition is not allowed
(1) 4^3 = 64
(2) 4P3 = 4*3*2 = 24
To find the number of three-digit numbers that can be formed using the digits 1, 2, 3, and 4, we can analyze the two scenarios separately.
1) Repetition is allowed:
In this case, each digit can be used repeatedly to form a three-digit number.
To form the first digit, we have four options: 1, 2, 3, and 4.
For the second and third digits, we also have four options each.
Using the multiplication principle, the total number of three-digit numbers can be formed with repetition allowed is:
4 (options for the first digit) x 4 (options for the second digit) x 4 (options for the third digit) = 64.
So, 64 three-digit numbers can be formed in the scenario where repetition is allowed.
2) Repetition is not allowed:
In this case, each digit can only be used once to form a three-digit number.
To form the first digit, we still have four options.
For the second digit, since repetition is not allowed, we have three options left after using the first digit.
For the third digit, we have two options left after using the first two digits.
Using the multiplication principle, the total number of three-digit numbers that can be formed with repetition not allowed is:
4 (options for the first digit) x 3 (options for the second digit) x 2 (options for the third digit) = 24.
So, 24 three-digit numbers can be formed in the scenario where repetition is not allowed.
To find the number of three-digit numbers that can be formed using the digits 1, 2, 3, and 4, we can consider two cases:
Case 1: Repetition is allowed
When repetition is allowed, each digit can be chosen from the set {1, 2, 3, 4}. Therefore, for each digit, there are 4 choices. Since we need to form a three-digit number, we have 4 choices for the hundreds place, 4 choices for the tens place, and 4 choices for the ones place. Therefore, the total number of three-digit numbers that can be formed with repetition allowed is 4 * 4 * 4 = 64.
Case 2: Repetition is not allowed
When repetition is not allowed, the first digit can be chosen from the set {1, 2, 3, 4}, which gives us 4 choices. After choosing the first digit, the second digit can be chosen from the remaining three digits, which gives us 3 choices. Finally, the third digit can be chosen from the remaining two digits, which gives us 2 choices. Therefore, the total number of three-digit numbers that can be formed without repetition is 4 * 3 * 2 = 24.
So, the answer to your question is:
(1) If repetition is allowed, there are 64 three-digit numbers that can be formed.
(2) If repetition is not allowed, there are 24 three-digit numbers that can be formed.