Three numbers are chosen at random from numbers 1 to 30. Write the pro

1.	Three numbers are chosen at random from numbers 1 to 30. Write the probability that the chosen numbers are consecutive.
Answer» Let E denote the event that the chosen numbers are consecutive. No of ways in which 3 numbers can be chosen out of 30 = ³⁰C₃ As we have to select 3 consecutive numbers, if we select 1 number other two are already selected. As 29,30 can’t be selected because if they are selected we won’t be able to get 3 consecutive numbers. ∴ number of ways in which 3 consecutive numbers can be selected = number of ways in which 1 number can be chosen out of numbers from 1 to 28 = ²⁸C₁ ways ∴ P(E) = \(\frac{28}{^{30}C_3}\) = \(\frac{28\times3\times2\times1}{30\times29\times28}\) = \(\frac{1}{145}\) Thus, P(E) = \(\frac{1}{145}\)

Three numbers are chosen at random from numbers 1 to 30. Write the probability that the chosen numbers are consecutive.

Answer»

Let E denote the event that the chosen numbers are consecutive.

No of ways in which 3 numbers can be chosen out of 30 = ³⁰C₃

As we have to select 3 consecutive numbers, if we select 1 number other two are already selected.

As 29,30 can’t be selected because if they are selected we won’t be able to get 3 consecutive numbers.

∴ number of ways in which 3 consecutive numbers can be selected = number of ways in which 1 number can be chosen out of numbers from 1 to 28 = ²⁸C₁ ways

∴ P(E) = \(\frac{28}{^{30}C_3}\) = \(\frac{28\times3\times2\times1}{30\times29\times28}\) = \(\frac{1}{145}\)

Thus, P(E) = \(\frac{1}{145}\)

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