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Discrete Math: Assignment I
January 9, 2007
1. Find the error in the following inductive proof showing that all the integers
in any set of integers are equal:
Base Case: Trivial!
Inductive Step: Suppose all sets of size n − 1 have all elements equal.
Now given a set S of size n consider the two sets S1 ,S2 of size n − 1 each
such that only the first element of S is missing from S1 while only the
last element is missing from S2 . Then by induction all elements of S1 are
equal and so are all elements of S2 hence all elements of S are equal to
this common value.
2. Use the equivalence principle to show that the number of partititions of
a set with n elements into j1 classes of size 1, j2 classes of size 2,..., jn
classes of size jn is
n!
Qn
ji
i=1 ((i!) ji !)
3. How many labellings are there for n-element sets with ji of the labels used
i times (1 ≤ i ≤ n)?
4. Three dice are thrown. Each dice can have a square with one of {1, 2, 3, 4, 5, 6}
facing up.
(a) How many different possibilities are there for the numbers facing up
with different colored dice?
(b) How many possibilities are there with two kinds of dice?
(c) How many different multisets of 3 numbers are possible?
(d) How many possibilities are there of getting distinct numbers in a
throw?
5. In how many ways can k identical pieces of candy be distributed among
n children if each child is to get at least j pieces?
6. What is the number of ways of obtaining the positive integer k as a sum
of a list of n
(a) non-negative integers?
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(b) positive integers?
7. Prove:
Pn
n−1
(a)
k=0 C(n, k) = n2
(b) * For n ≥ 0,
XX
C(n − i, j)C(n − j, i) = f2n+1
i≥0 j≥0
Here C(n, k) are the binomial coefficients while fn is the nth Fibonacci
number (defined as f0 = 1, f1 = 1 and fn = fn−1 + fn−2 ). Try to give a
combinatorial and an inductive proof. Can you think of any other proof
technique?
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