A ship carries 5 flags of different colours. A signal is made by hoisting one or more of them on a mast, one above another; two signals count as different if the flags used differ or if the order in which they hang differs. How many different signals can be made?

A signal that uses r flags can be made in $P$(5,r) ways, because the order up the mast matters. Add over every allowed r: $P(5,1)+P(5,2)+P(5,3)+P(5,4)+P(5,5)=5+20+60+120+120=325$. Trap 120 = 5! counts only the signals that use all five flags. Trap $31=2^5-1$ counts WHICH flags are hoisted but treats the mast as unordered, so it loses every rearrangement. Trap 205 leaves out the five-flag case worth 120; note that 205 + 120 = 325.