Our club has 20 members, 10 boys and 10 girls.  In how many ways can we choose a president and a vice-president if they must be of different gender?
I can do this by choosing the president first, and then the vice-president. If I choose a boy as the president, then I have 10 choices for him, and then 10 choices for the vice-president, who must be a girl. Similarly, if I choose a girl as the president, then I have 10 choices for her, and then 10 choices for the vice-president, who must be a boy. So, the total number of ways to choose a president and a vice-president of different genders is 10 times 10 plus 10 times 10, which is 200.

