MathDB

In a class there are n students with unequal heights.

<span class='latex-bold'>(a)</span>

Find the number of orderings of the students such that the shortest person is not at the front and the tallest person is not at the end.

<span class='latex-bold'>(b)</span>

Define the badness of an ordering as the maximum number

k

such that there are

k

many people with height greater than in front of a person. For example: the sequence

66, 61, 65, 64, 62, 70

has badness

3

since there are

3

numbers greater than

62

in front of it. Let

f_k(n)

denote the number of orderings of

n

with badness

k

. Find

f_k(n)

. (Hint: Consider

g_k(n)

as the number of orderings of n with badness less than or equal to

k

)

Problem Statement