Question

Number of non-empty subsets of $\{1,2,3,4,5,6,7,8\}$ having exactly $k$ elements and do not contain the element $k$ for some $k=1,2,\dots \dots \dots ..,8$ is

• A. 63
• B. 255
• C. 127
• D. 31

Answer 1

Answer: (C)

Single element subsets ${}^7{C}_1$ ( 1 cannot be taken $)$
Two element subsets ${}^7{C}_2$ ( 2 cannot be taken)
Three element subsets ${}^7{C}_3$ ( 3 cannot be taken)
Four element subsets ${}^7{C}_4(4$ cannot be taken $)$
Five element subsets ${}^7{C}_5$ ( 5 cannot be taken)
Six element subsets ${}^7{C}_6(6$ cannot be taken $)$
Seven element subsets ${}^7{C}_7$ ( 7 cannot be taken)
$\therefore$ Total number of non-empty subsets are
${}^7{C}_1+{}^7{C}_2+{}^7{C}_3+\dots \dots .+{}^7{C}_7=127$