At St. Xavier High School ten candidates C1, C2, ..., C10, run for senior class president.
(a) How many outcomes are possible where (i) there are no ties (that is, no two, or more, candidates receive the same number of votes? (ii) ties are permitted? [Here we may have an outcome such as {C2, C3, C7}, {C1, C4, C9, C10}, {C5}, {C6, C8}, where C2, C3, C7 tie for first place, C1, C4, C9, C10 tie for fourth place, C5 is in eighth place, and C6, C8 are tied for ninth place.] (iii) three candidates tie for first place (and other ties are permitted)?
(b) How many of the outcomes in section (iii) of part (a) have C3 as one of the first-place candidates?
(c) How many outcomes have C3 in first place (alone, or tied with others)?