Ordered compositions of a number

Text

In how many ways can $7$ be written as an ordered sum of positive integers? (For example $7$, $\ 3+4$, $\ 4+3$, $\ 1+1+5$ count as different if the order of the terms changes.)

Solution

Picture $7$ as seven units in a row: $1\ 1\ 1\ 1\ 1\ 1\ 1$. In the $7-1=6$ gaps between units we may or may not place a "$+$" sign: each independent choice $\{$place, do not place$\}$ gives a different ordered sum. Hence $N = 2^{7-1} = 2^6 = \boxed{64}.$