How many four-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5fi
a. no digit may be repeated in a number?
b. the numbers are to be even and no digits are repeated?
c. the numbers are less than 4500, and no digits are to be repeated? d. the numbers are to be odd and no digits are repeated?
a. Since no digit may be repeated, there are 6 choices for the first digit, 5 choices for the second digit, 4 choices for the third digit, and 3 choices for the fourth digit. Therefore, the total number of four-digit numbers that can be formed is 6 x 5 x 4 x 3 = 360.
b. Since the numbers are to be even and no digits are repeated, the last digit must be 0, 2, or 4. For the first digit, there are 3 choices (2,3,4) and for the second and third digits, there are 4 choices each. Therefore, the total number of four-digit even numbers with no repeated digits is 3 x 4 x 4 x 1 = 48.
c. Since the numbers are less than 4500 and no digits are repeated, the first digit must be 1, 2, or 3. For the first digit, there are 3 choices, and for the second, third, and fourth digits, there are 4 choices each. Therefore, the total number of four-digit numbers less than 4500 with no repeated digits is 3 x 4 x 4 x 4 = 192.
d. Since the numbers are to be odd and no digits are repeated, the last digit must be 1, 3, or 5. For the first digit, there are 3 choices (1,2,3), and for the second, third, and fourth digits, there are 4 choices each. Therefore, the total number of four-digit odd numbers with no repeated digits is 3 x 4 x 4 x 3 = 144.