Question

How many odd numbers of five distinct digits can be formed with the digits 0,1?

Answer 1

 will warn you about naming things "permutations" but then treating them as different objects. The phrasing of the question in the body is correct and unambiguous ("how many odd numbers of 5 digits..."), but the concept of an "odd permutation" is distinctly different than that of an odd number. For example, 52340 is an odd permutationof the digits when treated as a permutation of {0,2,3,4,5}{0,2,3,4,5} despite being an even number. If in doubt of whether something should be referred to as a permutation, "arrangement" is a more generic term which is usually valid and often more appropriate.