# Question

Let Hk(n) be the number of vectors x1, . . ., xk for which each xi is a positive integer satisfying 1 ≤ xi ≤ n and x1 ≤ x2 ≤ . . . ≤ xk.

(a) Without any computations, argue that

H1(n) = n

How many vectors are there in which xk = j?

(b) Use the preceding recursion to compute H3(5).

First compute H2(n) for n = 1, 2, 3, 4, 5.

(a) Without any computations, argue that

H1(n) = n

How many vectors are there in which xk = j?

(b) Use the preceding recursion to compute H3(5).

First compute H2(n) for n = 1, 2, 3, 4, 5.

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