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# Prove the following theorem: The total number of subsets of a finite set containing n elements is 2n.

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## To prove the above statement we have to use the combinations and binomial theorem. so first I will take an example then we can generalize it. Consider a set a containing 3 elements. A= {1,2,3} First let's select 0 elements from 3 elements. That's no elements so that we will get the null set. No. of selection of 0 elements from 3 elements is 3C0 = 1 Then we select 1 element from 3.ie; {1},{2},{3} No. of selection of 1 element from 3 elements is 3C1=3 Select 2 elements from 2. i.e.; {1,2} {1,3} {2,3}. No. of selection of 2 elements from 3 elements is 3C2=3 Select 3 elements from three. i.e.; {1,2,3}. No. of selection of 3 elements from 3 elements is 3C3 = 1 So total no. of subsets of the above finite set is 3C+3C1+3C2+3C3 = 1+3+3+1 = 8 = 23 So when we generalize the case in the case of subset containing n elements the total no. of subsets is nC0 + nC1 + nC2 + nC3 + nC4 + ..................... + nCn In binomial theorem we have studied the expansion of (1+x )n = nC0 + nC1 x + nC2 x2 + nC3 x3 +nC4 x4 + .......... + nCn xn Substitute 1 for x So in the L.H.S we get (1+1)n = 2n In R.H.S we get nC0+nC1+nC2 +nC3 +nC4 +..........+nCn So nC0 + nC1 + nC2 + nC3 + nC4 + .......... + nCn = 2n Hence the total no. of subsets of a finite set containing n elements is 2n  Suggest Corrections  10      Similar questions
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