QJEEM23Made5f: JEE Main 2023 Mathematics - Class 11 JEEM23Made5f Solution: Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty…

Elements of the type $3 \mathrm{k}=3$

Elements of the type $3 \mathrm{k}+1=1,7,9$

Elements of the type $3 \mathrm{k}+2=2,5,11$

Subsets containing one element $S_1=1$

Subsets containing two elements

$$S_2={}^3{C}_1\times {}^3{C}_1=9$$

Subsets containing three elements

$${\mathrm{S}}_3={}^3{\mathrm{C}}_1\times {}^3{\mathrm{C}}_1+1+1=11$$

Subsets containing four elements

$${\mathrm{S}}_4={}^3{\mathrm{C}}_3+{}^3{\mathrm{C}}_3+{}^3{\mathrm{C}}_2\times {}^3{\mathrm{C}}_2=11$$

Subsets containing five elements

$${\mathrm{S}}_5={}^3{\mathrm{C}}_2\times {}^3{\mathrm{C}}_2\times 1=9$$

Subsets containing six elements $\mathrm{S}_6=1$

Subsets containing seven elements $\mathrm{S}_7=1$

$$\Rightarrow \text{ sum }=43$$