(a) No restrictions: 10000-99999, 90000 different numbers (if zero disallowed as first digit) or 00000-99999, 100000 different numbers (if zero allowed as first digit).
(b) No repeat digits: There are 252 numbers formed from the digits 0-9 (select 5 out of 10), and these include those beginning with zero. If zero is the first digit, then there 126 numbers 4-digit numbers to fill in the remaining digits (select 4 out of 9). So we subtract 126 from 252=126 5-digit numbers, if zero is disallowed as first digit.
(c) If zero is disallowed as first digit, 10000-99998 is the range for even numbers: divide by 2: 5000-49999, which is 45000 numbers, including repeat digits. If zero is allowed, 00000-99998 is the range: divide by 2: 0-499999, 50000 numbers.
(d) First digit is 2: the remaining 4 digits can be 0000-9999 (range is 20000-29999), 10000 numbers.
(e) Non-repeating odd digits. 1, 3, 5, 7, 9. There are 120 ways of arranging these digits, so there are 120 numbers.