A five-member advisory panel is to be picked from 9 women and 6 men so that it holds exactly 3 women, and then one of the five chosen is to be named chairperson and a different one secretary. In how many ways can the whole thing be done?

Pick the panel first: 3 women from 9 in $C(9,3)=84$ ways and, since the panel has five places, 2 men from 6 in $C(6,2)=15$ ways — that is $84\times 15=1260$ possible panels. Then fill the posts: they are different posts, so the chairperson can be any of the 5 and the secretary any of the remaining 4, giving $5\times 4=20$. Total = $1260\times 20=25200$. Trap 1260 stops at the panel and never fills the posts. Trap $12600=1260\times C(5,2)$ picks the two office-bearers as an unordered pair and so loses the distinction between chairperson and secretary. Trap $6300=1260\times 5$ names only the chairperson.