Let S = 1 , 2 , 3 , 5 , 7 , 10 , 11 . The number of non-empty subsets of S that have the sum of all elements…

Given set is

$S=\left\{1,2,3,5,7,10,11\right\}$

Elements of the type $3k$ is $3$.

Elements of the type $3k+1$ are $\left\{1,7,9\right\}$.

Elements of the type $3k+2$ are $\left\{2,5,11\right\}$

Subset containing one element $S_1=1$

Subsets containing two elements $S_2$ 

$={}^{3}C_1\times {}^{3}C_1=9$

Subsets containing three elements $S_3$

$={}^{3}C_1\times {}^{3}C_1+1+1=11$

Subsets containing four elements $S_4$

$={}^{3}C_3+{}^{3}C_3+{}^{3}C_2\times {}^{3}C_2=11$

Subsets containing five elements $S_5$

$={}^{3}C_2\times {}^{3}C_2\times 1=9$

Subsets containing six elements $S_6=1$

Subsets containing seven elements $S_7=1$

Required number

$=1+9+11+11+9+1+1$

$=43$