How many 3-digit numbers can be formed from the four
integers 0, 1, 2, 3, if;
a) no digit is repeated,
b) Repetition is allowed,
c) The number is even and repetition is not allowed,
d) The number is odd and repetition is not allowed.
To find the number of 3-digit numbers that can be formed from a given set of integers, we can use the concept of permutations and combinations. Let's break down each case step by step:
a) When no digit is repeated, we have to choose 3 digits from the given set of 4 integers (0, 1, 2, 3). We can use the formula for combinations (nCr) to calculate the number of ways to choose the digits without repetition. In this case, the number of 3-digit numbers is calculated as:
Number of 3-digit numbers = 4C3 = 4! / (3! * (4-3)!) = 4
So, there are 4 different 3-digit numbers that can be formed when no digit is repeated.
b) When repetition is allowed, we can select any digit from the given set of 4 integers (0, 1, 2, 3) for each of the three positions. In this case, the number of 3-digit numbers is calculated as:
Number of 3-digit numbers = 4 * 4 * 4 = 4^3 = 64
So, there are 64 different 3-digit numbers that can be formed when repetition is allowed.
c) When the number is even and repetition is not allowed, the unit digit must be 0, 2, or 4. The first digit (hundreds place) can be chosen in 3 ways (excluding 0). The second digit (tens place) can be chosen in 3 ways (excluding the digit already chosen for the hundreds place). The unit digit (ones place) has 2 choices (0 or 2). In this case, the number of 3-digit even numbers is calculated as:
Number of 3-digit even numbers = 3 * 3 * 2 = 18
So, there are 18 different 3-digit even numbers that can be formed when even numbers are allowed, and repetition is not allowed.
d) When the number is odd and repetition is not allowed, the unit digit must be 1 or 3. The first digit (hundreds place) can be chosen in 3 ways. The second digit (tens place) can be chosen in 2 ways (excluding the digit already chosen for the hundreds place). The unit digit (ones place) has 2 choices (1 or 3). In this case, the number of 3-digit odd numbers is calculated as:
Number of 3-digit odd numbers = 3 * 2 * 2 = 12
So, there are 12 different 3-digit odd numbers that can be formed when odd numbers are allowed, and repetition is not allowed.