First slide

Introduction to probability

Question

The numbers 1,2,3, ..., n are arrange in a random order. The probability that the digits 1, 2, 3, . . ., k (k < n) appear as neighbors in that order is

Moderate

A

$\frac{1}{n!}$
B

$\frac{k!}{n!}$
C

$\frac{(n - k)!}{n!}$
D

$\frac{(n - k + 1)!}{n!}$

Solution

The number of ways of arranging n numbers is n! In each order obtained, we must now arrange the digits 1, 2, ..., k as group and the n - k remaining digits. This can be done in (n - k + l)! ways. Therefore, the probability for the required event is $\frac{(n - k + 1)!}{n!}$

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