Let S = \ 1, 2, 3, 5, 7, 10, 11\ . The number of non-empty subsets...

Given set is

$S = \left\{1, 2, 3, 5, 7, 10, 11\right\}</p><p>Elements$ of the type3k$is$3$.</p><p>Elements of the type$3k + 1$are \left\{1, 7, 9\right\}.</p><p>Elements$ of the type3k + 2$are \left\{2, 5, 11\right\}</p><p>Subset containing one element${S}_{1} = 1$</p><p>Subsets containing two elements${S}_{2}$</p><p>$= C1 3 \times C1 3 = 9$</p><p>Subsets containing three elements${S}_{3}$</p><p>$= C1 3 \times C1 3 + 1 + 1 = 11$</p><p>Subsets containing four elements${S}_{4}$</p><p>$= C3 3 + C3 3 + C2 3 \times C2 3 = 11$</p><p>Subsets containing five elements${S}_{5}$</p><p>$= C2 3 \times C2 3 \times 1 = 9$</p><p>Subsets containing six elements${S}_{6} = 1$</p><p>Subsets containing seven elements${S}_{7} = 1$</p><p>Required number</p><p>$= 1 + 9 + 11 + 11 + 9 + 1 + 1$</p><p>$= 43$</p>$