Multiple choice

If $ \displaystyle n $ integers taken at random are multiplied together, the chance (probability) that the last digit of the product is 2,4,6 or 8 is

  1. $ \displaystyle \dfrac{5^{n}-4^{n}}{10^{n}} $
  2. $ \displaystyle \dfrac{2^{n}}{5^{n}} $
  3. $ \displaystyle \dfrac{4^{n}-2^{n}}{5^{n}} $

  4. None of these
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C Correct answer
Explanation

The last digit of a product is 2, 4, 6, or 8 if the product is even but not a multiple of 5. The probability that a single integer is not a multiple of 5 is 4/5, and the probability that the product of n integers is not a multiple of 5 is (4/5)^n. The probability that the product is even is 1 - (1/2)^n. Combining these, the probability that the product is even and not a multiple of 5 is (4/5)^n - (2/5)^n, which simplifies to (4^n - 2^n) / 5^n.

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