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Classical Definition of Probability

Two numbers x...

Question

Two numbers x and y are chosen at random (without replacement) from amongst the numbers 1, 2, 3,……, 2004. The probability that $(x^{3} + y^{3})$ is divisible by 3 is $\frac{p}{q}$ (p and q are co-prime). Find the value of (p + q).___

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Solution

Let $E_{1} = 1, 4, 7$ , ...... (n each) [3n = 2004]

$E_{2} = 2, 5, 8,$ ...... (n each)

$E_{3} = 3, 6, 9,$ ...... (n each)

x and y belong to choosing one from $E_{1} and E_{2}, or choosing two out of E_{3}$

so, required probability $= \frac{^{n} C_{1}^{n} C_{1} +^{n} C_{2}}{^{3 n} C_{2}} = \frac{1}{3}$

$∴ \frac{p}{q} = \frac{1}{3} \Rightarrow p + q = 4$

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