Question

Let n be a natural number. We choose a sequence i 1 ,…,i n of numbers between 1 and n at random.

(a) What is the probability that the sequence contains exactly n−2 integers?
(b) What is the probability that the sequence contains exactly n−3 integers?

Answer 1

The probability of exactly n-2 integers in the sequence is n*(n-1)/(2*n^n), and the probability of exactly n-3 integers is n*(n-1)*(n-2)/(6*n^n).

Step-by-step explanation:

a. Probability of exactly n-2 integers:

The total number of possible sequences is n^n, as each number in the sequence can be any number from 1 to n.

The number of sequences that contain exactly n-2 integers is nCn-2 (n choose n-2), which can be calculated as n*(n-1)/2.

Therefore, the probability is n*(n-1)/(2*n^n).

b. Probability of exactly n-3 integers:

The number of sequences that contain exactly n-3 integers is nCn-3 (n choose n-3), which can be calculated as n*(n-1)*(n-2)/6.

Therefore, the probability is n*(n-1)*(n-2)/(6*n^n).