The number of subsets of \{1,2, \ldots .99\}{1,2,….99} containing at least 50 elements is
Answers
Given : A Set { 1 , 2 , 3 ........... , 99 }
To find : number of subsets containing at least 50 elements
Solution:
{ 1 , 2 , 3 ........... , 99 }
Total Elements = 99
subsets containing at least 50 elements
=> 50 elements , 51 elements ................................. , 99 elements
number of subsets containing 50 elements = ⁹⁹C₅₀
number of subsets containing 50 elements = ⁹⁹C₅₁
....
....
number of subsets containing 99 elements = ⁹⁹C₉₉
number of subsets containing at least 50 elements = ⁹⁹C₅₀ + ⁹⁹C₅₁ ............. + ⁹⁹C₉₉
⁹⁹C₅₀ + ⁹⁹C₅₁ ............. + ⁹⁹C₉₉
= 2 ( ⁹⁹C₅₀ + ⁹⁹C₅₁ ............. + ⁹⁹C₉₉ )/2
= (⁹⁹C₉₉+...... + ⁹⁹C₅₁ + ⁹⁹C₅₀ + ⁹⁹C₅₀ + ⁹⁹C₅₁ ............. + ⁹⁹C₉₉ )/2
ⁿCₓ = ⁿCₙ₋ₓ
= (⁹⁹C₀ +... .......+ ⁹⁹C₄₈ + ⁹⁹C₄₉ + ⁹⁹C₅₀ + ⁹⁹C₅₁ ............. + ⁹⁹C₉₉ )/2
= 2⁹⁹/2
= 2⁹⁸
2⁹⁸ is number of subsets containing at least 50 elements
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