Question

Find the total number of subsets of a set with
[Hint: ⁿC₀ + ⁿC₁ + ⁿC₂ + ... + ⁿCn = 2ⁿ] 5 elements

Answer 1

Subsets with 5 elements:

Number of subsets with no element = 5C₀

Number of subsets with one element = 5C₁

Number of subsets with 2 elements = 5 C₂

Number of subsets with 3 elements = 5C₃

Number of subjects with 4 elements = 5C₄

Number of subsets with 5 elements = 5C₅

Total number of subjects

= ⁵C₀ + ⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅

= `1 + (5!)/(1!(5 - 1)!) + (5!)/(2!(5 - 2)!) + (5!)/(3!(5 - 3)!) + (5!)/(4!(5 - 4)!) + 1`

= `1 + (5!)/(4!) + (5!)/(2!  3!) + (5!)/(3!  2!) + (5!)/(4!) + 1`

= `1 + (5 xx 4!)/(4!) + (5 xx 4 xx 3!)/(2! xx 3!) + (5 xx 4 xx 3!)/(3! xx 2!) + (5 xx 4!)/(4!) + 1`

= `1 + 5 + (5 xx 4)/(2 xx 1) + (5 xx 4)/(2 xx 1) + 5 + 1`

= 6 + 10 + 10 + 6

= 32