How many different five-digit number license plates can be made if
i. the first digit cannot be zero, and the repetition of digits is not allowed,
ii. the first-digit cannot be zero, but the repetition of digits is allowed?
(i) Given: Five-digit number is required in which the first digit cannot be zero, and the repetition of digits is not allowed.
Assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is 9C1
Similarly, the second box can be filled with one of the nine available digits, so the possibility is 9C1
the third box can be filled with one of the eight available digits, so the possibility is 8C1
the fourth box can be filled with one of the seven available digits, so the possibility is 7C1
the fifth box can be filled with one of the six available digits, so the possibility is 6C1
Hence, the number of total possible outcomes is 9C1 × 9C1 × 8C1 × 7C1 × 6C1 = 9 × 9 × 8 × 7 × 6 = 27216
(ii) Given: Five-digit number is required in which the first digit cannot be zero, and the repetition of digits is allowed
assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is 9C1
Similarly, the second box can be filled with one of the ten available digits, so the possibility is 10C1
the third box can be filled with one of the ten available digits, so the possibility is 10C1
the fourth box can be filled with one of the ten available digits, so the possibility is 10C1
the fifth box can be filled with one of the ten available digits, so the possibility is 10C1
Hence, the number of total possible outcomes is 9C1 × 10C1 × 10C1 × 10C1 × 10C1 = 9 × 10 × 10 × 10 × 10 = 90000