A box contains coupons labeled 1,2,3,....n. A coupon is picked at random and the number x is noted. The coupon is put back into the box and a new coupon is picked at random. The new number is y. Then the probability that one of the numbers x, y divides the other is ([.] is greatest integer function, and the summations are from k = 1 to n)
A. 1/2
B. 1n2Σ[nk]
C. -1n + 1n2Σ[nk]
D. -1n + 2n2Σ[nk]
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2 Answers
2305You can choose x in n ways and y in n ways.
Let y>x,
Number of multiples of x within 1 to n is = \lfloor\frac{n}{x}\rfloor
Hence probability that x divides y = \frac{\lfloor\frac{n}{x}\rfloor}{n^2}
x ranges from 1 to n,
Therefore the required probability =
.\qquad\frac{1}{n^2}\sum\lfloor\frac{n}{x}\rfloor
So I think the answer should be (B).
