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 $k =1,2, \ldots \ldots \ldots . ., 8$ 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
${ }^7 C _1+{ }^7 C _2+{ }^7 C _3+\ldots \ldots .+{ }^7 C _7=127$