You have 2n numbers such where each number i is repeated twice (1 ≤ i ≤ n). Given a random permutation of these 2n numbers what is the expected number of adjacent pairs that have the same value. For example, if n = 5, then the permutation 1 2 2 3 4 5 5 4 3 1 has two adjacent pairs that have the same value.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
You have 2n numbers such where each number i is repeated twice (1 ≤ i ≤ n). Given a
random permutation of these 2n numbers what is the expected number of adjacent pairs that have
the same value.
For example, if n = 5, then the permutation 1 2 2 3 4 5 5 4 3 1 has two adjacent pairs that have the
same value.
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