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Find the total number of subsets of a set with

[Hint: ^{n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + ... + ^{n}Cn = 2^{n}] 5 elements

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#### Solution

Subsets with 5 elements:

Number of subsets with no element = 5C_{0}

Number of subsets with one element = 5C_{1}

Number of subsets with 2 elements = 5 C_{2}

Number of subsets with 3 elements = 5C_{3}

Number of subjects with 4 elements = 5C_{4}

Number of subsets with 5 elements = 5C_{5}

Total number of subjects

= ^{5}C_{0} + ^{5}C_{1} + ^{5}C_{2} + ^{5}C_{3} + ^{5}C_{4} + ^{5}C_{5}

= `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

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