A quiz panel of 5 is to be picked from 4 teachers and 7 students of a school. In how many ways can the panel be picked if it must include at least one teacher?

Count everything and take away the banned case. All panels: C(11,5) = 462. Panels with no teacher at all: C(7,5) = 21. So panels with at least one teacher = 462 − 21 = 441. Trap 840 = 4 × C(10,4) tries to 'reserve one teacher first, then pick any 4 from the rest'; that counts a panel holding two teachers twice over, once for each teacher reserved. Trap 462 ignores the condition and trap 21 reports the banned case itself.