Question

Assertion :Let A be any set with 10 distinct elements then number of non empty subsets of A are 1023. Reason: The number of ways of selecting one or more items from a group of n distinct items is $2^n-1.$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R) is false,
• D. (A)is false but (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

Out of n items,1 can be selected by ${}^n{C}_1$ ways,
Two items can be selected by ${}^n{C}_2$ ways, three can be selected by ${}^n{C}_3$ ways so on
$\therefore$ Total number of ways of selecting at least one item
${=}^n{C}_1{+}^n{C}_2{+}^n{C}_3+.......{.}^n{C}_n{=}^n{C}_0{+}^n{C}_1{+}^n{C}_2+.........{+}^n{C}_n{-}^n{C}_0$
$=2^n{-}^n{C}_0$ $=2^n-1$ For
$n=10$ the number of ways are given by $2^{10}-1=1024-1=1023$