Three numbers are chosen without replacement from { 1 , 2 , 2 2 , … , 2 n } . Which formula gives the probability that they form an increasing geometric progression?

Count three-term arithmetic progressions among exponents 0 , … , n . For odd n the ratio to ( 3 n + 1 ) is 3/ ( 2 n ) ; for even n it is 3 n / [ 2 ( n 2 − 1 ) ] .

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MathematicsProbabilityGeometrical ProbabilityMedium2 minQB

MathematicsMediummultiple choice

Three numbers are chosen without replacement from ${1, 2, 2^{2}, \dots, 2^{n}}$ . Which formula gives the probability that they form an increasing geometric progression?

Options:(select one or more)

Answer:

A, C

Solution:

Count three-term arithmetic progressions among exponents $0, \dots, n$ . For odd $n$ the ratio to $(3 n + 1)$ is $3/ (2 n)$ ; for even $n$ it is $3 n / [2 (n^{2} - 1)]$ .

Stream:JEESubject:MathematicsTopic:ProbabilitySubtopic:Geometrical Probability

⏱ 2mℹ️ Source: QB

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