Question

Find the total number of subsets of a set with
[Hint: ⁿC₀ + ⁿC₁ + ⁿC₂ + ... + ⁿC_n = 2ⁿ] 4 elements

Answer 1

Subsets with 4 elements

Number of subsets with no element = 4C_o

Number of subsets with one element = 4C₁

Number of subsets with two elements = 4C₂

Number of subsets with three elements = 4C₃

Number of subsets with four elements = 4C₄

∴ Total number of subsets

= ⁴C₀ + ⁴C₁ + ⁴C₂ +  ⁴C₃ + ⁴C₄ 

= `(4!)/(0!(4 - 0)!) + (4!)/(1!(4 - 1)!) + (4!)/(2!(4 - 2)!) + (4!)/(3!(4 - 3)!) + (4!)/(4!(4 - 4)!)`

= `(4!)/(4!) + (4!)/(3!)  (4!)/(2! xx 2!) + (4!)/(3! xx 1!) + (4!)/(4! xx 0!)`

= `1 + (4 xx 3!)/(3!) + (4 xx 3  xx 2!)/(2 xx 1 xx 2!) + (4 xx 3!)/(3!) + (4!)/(4!)`

= 1 + 4 + 6 + 4 + 1

= 16