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Consider a set, S, consisting of n elements.
Then the number of subsets of S containing r objects is equal to the number of ways of choosing r elements out of n = nCr.Therefore
nC0 + nC1 + nC2 + ... + nCn is the total number of subsets of S.
Now consider each subset of S. Each element of S can either be in the subset or not in the subset. So for each element there are two choices. Therefore, n elements give 2^n possible subsets.
Since these are two different approaches to the total number of subsets of S,
nC0 + nC1 + nC2 + ... + nCn = 2^n.
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What is the probability that there will be 93.75 percent or more HEADS in n tosses?
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What is the difference between nc2 and nc4?
The main difference between nC2 and nC4 is the number of elements being chosen. nC2 represents the number of ways to choose 2 elements from a set of n elements, while nC4 represents the number of ways to choose 4 elements from a set of n elements. In general, nC4 will be a larger number compared to nC2 for the same value of n.
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1 plus r raised to power n equals?
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On a circular island we build n straight dams going from sea to sea so that every two intersect but no three go through the same point Use Euler's Formula to determine how many parts we get?
Consider following figure with n = 4. Clearly, number of vertices (v) = 14, number of edges (e) = 24 and number of faces (f) = 12. According to Euler's formula, v + f = e + 2 i.e. v + f = 14 + 12 = 26 = e + 2. In general we have, v = 2n + nC2 and e = 2n + n2. Hence, using Euler's formula, f = e + 2 - v = 2n + n2 + 2 - 2n + nC2 After simiplication we have,
What is binomial expansion theorem?
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What is the probability that 2 out of 15 students will have the same birthday?
P( 2 equal birthday in group of 15 ) ≈ 22.3%The probability that 2 persons and only 2 persons in a random group of n persons can be calculated using the following expression*:P(2 equal bd in group of n persons) = nC2(1-1/1461)(4/1461)Π2n-1[1-4(i-1)/1461] + nC2(1/1461)2Π2n-1[1-(4i-3)/1461]with n=15, 30C2 = 105,P(2 eq bd in group of 15) = 105(1-1/1461)(4/1461)[1-4/1461][1-8/1461][1-12/1461]∙∙∙[1-52/1461]P(...) = 0.223263549... ≈ 22.3%*You find this expression discussed in question "What is the probability that in a room of 8 people 2 have the same birthday ?". A simpler form of this expression that neglects February 29 of the leap day that gives a goodapproximation is:P(2 eq bd in group of n persons) = nC2(1/365)Π1n-1[1-(i-1)/365]for n=15, this expression gives P(...) = 0.214918798... ≈ 21.5%
How many diagonals do hexagons have?
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