Combinatorics Through Guided Discovery, 2004a

70 3. Distribution Problems
⇒
Problem 173. Show that the number of partitions of 7 into 3 parts equals
the number of partitions of 10 into three distinct parts. (h)
⇒ ·
Problem 174. There is a relationship between P(k, n) and Q(m, n) for some
other number m. Find the number m that gives you the nicest possible
relationship. (h)
· Problem 175. Find a recurrence that expresses Q(k, n) as a sum of Q(k −
n, m) for appropriate values of m. (h)
⇒∗
Problem 176. Show that the number of partitions of k into distinct parts
equals the number of partitions of k into odd parts. (h)
⇒∗ Problem 177. Euler showed that if k 3j2 +j
2
, then the number of partitions
of k into an even number of distinct parts is the same as the number of
partitions of k into an odd number of distinct parts. Prove this, and in the
exceptional case find out how the two numbers relate to each other. (h)
3.4 Supplementary Problems
1. Answer each of the following questions with n k , k n , n!, k!, ( n k ), (k n ), nk , k n , n k ,
k n , ( n+k−1
k
), ( n+k−1
n
), ( n−1 ), (k−1 ),
k−1 n−1
or “none of the above".
(a) In how many ways may we pass out k identical pieces of candy to n children?
(b) In how many ways may we pass out k distinct pieces of candy to n children?
(c) In how many ways may we pass out k identical pieces of candy to n children
so that each gets at most one? (Assume k ≤ n.)
(d) In how many ways may we pass out k distinct pieces of candy to n children
so that each gets at most one? (Assume k ≤ n.)
(e) In how many ways may we pass out k distinct pieces of candy to n children
so that each gets at least one? (Assume k ≥ n.)
(f) In how many ways may we pass out k identical pieces of candy to n children
so that each gets at least one? (Assume k ≥ n.)
2. The neighborhood betterment committee has been given r trees to distribute to
s families living along one side of a street.