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for-A-1-2-3-4-5-6-7-compute-the-number-of-a-Subsets-of-A-b-Nonempty-subsets-of-A-c-proper-subsets-of-A-d-Non-empty-proper-subset-of-A-e-Subsets-of-A-containing-three-element-f-Subs

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Question Number 102357 by Cynosure last updated on 08/Jul/20
for A={1,2,3,4,5,6,7},compute the number of: (a) Subsets of A. (b) Nonempty subsets of A. (c) proper subsets of A. (d) Non empty proper subset of A. (e) Subsets of A containing three element. (f) Subsets of A containing 1,2. (g) Proper subsets of A containing 1,2. (h) Subset of A with an even number of element. (i) Subset of A with an odd number of element. (j) Subsets of A with an odd number of elements, including the element 3.
$${for}\:{A}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7}\right\},{compute}\:{the}\:{number}\:{of}: \\ $$$$\left({a}\right)\:{Subsets}\:{of}\:{A}. \\ $$$$\left({b}\right)\:{Nonempty}\:{subsets}\:{of}\:{A}. \\ $$$$\left({c}\right)\:{proper}\:{subsets}\:{of}\:{A}. \\ $$$$\left({d}\right)\:{Non}\:{empty}\:{proper}\:{subset}\:{of}\:{A}. \\ $$$$\left({e}\right)\:{Subsets}\:{of}\:{A}\:{containing}\:{three}\:{element}. \\ $$$$\left({f}\right)\:{Subsets}\:{of}\:{A}\:{containing}\:\mathrm{1},\mathrm{2}. \\ $$$$\left({g}\right)\:{Proper}\:{subsets}\:{of}\:{A}\:{containing}\:\mathrm{1},\mathrm{2}. \\ $$$$\left({h}\right)\:{Subset}\:{of}\:{A}\:{with}\:{an}\:{even}\:{number}\:{of}\:{element}. \\ $$$$\left({i}\right)\:{Subset}\:{of}\:{A}\:{with}\:{an}\:{odd}\:{number}\:{of}\:{element}. \\ $$$$\left({j}\right)\:{Subsets}\:{of}\:{A}\:{with}\:{an}\:{odd}\:{number}\:{of}\:{elements},\:{including}\:{the}\:{element}\:\mathrm{3}. \\ $$
Answered by bobhans last updated on 09/Jul/20
(a) 2^7 = 128 (b) 2^7 −1=127 (e) C_3 ^7 = ((7×6×5)/(3×2×1)) = 35
$$\left({a}\right)\:\mathrm{2}^{\mathrm{7}} \:=\:\mathrm{128} \\ $$$$\left({b}\right)\:\mathrm{2}^{\mathrm{7}} −\mathrm{1}=\mathrm{127} \\ $$$$\left({e}\right)\:{C}_{\mathrm{3}} ^{\mathrm{7}} \:=\:\frac{\mathrm{7}×\mathrm{6}×\mathrm{5}}{\mathrm{3}×\mathrm{2}×\mathrm{1}}\:=\:\mathrm{35} \\ $$$$ \\ $$
Commented by Cynosure last updated on 08/Jul/20
(d) should not be 35 sir
$$\left({d}\right)\:{should}\:{not}\:{be}\:\mathrm{35}\:{sir} \\ $$
Commented by Rasheed.Sindhi last updated on 08/Jul/20
(d)Nonempty proper subsets 128−2=126
$$\left({d}\right){Nonempty}\:{proper}\:{subsets} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{128}−\mathrm{2}=\mathrm{126} \\ $$
Commented by Rasheed.Sindhi last updated on 08/Jul/20
Typo in 3rd line (e) C_3 ^7 = ((7×6×5)/(3×2×1)) = 35
$$\mathcal{T}{ypo}\:{in}\:\mathrm{3}{rd}\:{line} \\ $$$$\left({e}\right)\:{C}_{\mathrm{3}} ^{\mathrm{7}} \:=\:\frac{\mathrm{7}×\mathrm{6}×\mathrm{5}}{\mathrm{3}×\mathrm{2}×\mathrm{1}}\:=\:\mathrm{35} \\ $$
Commented by bobhans last updated on 09/Jul/20
yes sir. thank you
$${yes}\:{sir}.\:{thank}\:{you} \\ $$
Answered by bemath last updated on 09/Jul/20
(f){1,2} ⇒1 {1,2,_}⇒C_1 ^5 = 5 {1,2,_,_}⇒C_2 ^5 = 10 {1,2,_,_,_}⇒C_3 ^5 = 10 {1,2,_,_,_,_}⇒C_4 ^5 = 5 {1,2,_,_,_,_,_}⇒C_5 ^5 = 1 totally 32
$$\left({f}\right)\left\{\mathrm{1},\mathrm{2}\right\}\:\Rightarrow\mathrm{1} \\ $$$$\left\{\mathrm{1},\mathrm{2},\_\right\}\Rightarrow{C}_{\mathrm{1}} ^{\mathrm{5}} \:=\:\mathrm{5} \\ $$$$\left\{\mathrm{1},\mathrm{2},\_,\_\right\}\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{5}} \:=\:\mathrm{10} \\ $$$$\left\{\mathrm{1},\mathrm{2},\_,\_,\_\right\}\Rightarrow{C}_{\mathrm{3}} ^{\mathrm{5}} \:=\:\mathrm{10} \\ $$$$\left\{\mathrm{1},\mathrm{2},\_,\_,\_,\_\right\}\Rightarrow{C}_{\mathrm{4}} ^{\mathrm{5}} \:=\:\mathrm{5} \\ $$$$\left\{\mathrm{1},\mathrm{2},\_,\_,\_,\_,\_\right\}\Rightarrow{C}_{\mathrm{5}} ^{\mathrm{5}} \:=\:\mathrm{1} \\ $$$${totally}\:\mathrm{32}\: \\ $$

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