Partitions of 8 into distinct parts

Text

In how many ways can $8$ be written as a sum of positive integers all different from one another (order does not matter)?

Solution

List them: $8;\ \ 7{+}1;\ \ 6{+}2;\ \ 5{+}3;\ \ 5{+}2{+}1;\ \ 4{+}3{+}1.$ (Four distinct parts are impossible, since $1+2+3+4=10>8$.) There are $\boxed{6}$ partitions.