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Let X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission is x number of colleges. It is given that :

where k is a positive constant. Find the value of k. Also, find the probability that you will get admission in 

(i) Exactly one college 

(ii) At most 2 colleges 

(iii) At least 2 colleges.

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We know that, sum of all the probabilities of  distribution is always 1.

(i) P(getting admission in exactly one college)

= P(X = 1) =1/8 (Using (i))

(ii) P (getting admission in at most 2 colleges)

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