MATH2302 MATH7308

The last question has two versions. One is for MATH2302, the other is for MATH7308. Submit only
the one corresponding to your course code!
1. (10 marks total) We would like to buy ten balloons for a birthday party. We go to a store that
sells individual balloons, in six colours: red, orange, yellow, green, blue, and purple.
(a) (3 marks) How many ways can we choose the colour of the balloons if the store has at least
10 of each colour?
(b) (3 marks) How many different choices of balloons can we make if the store has at least 10
of each colour, but we would like to make sure to buy at least 1 yellow and at least 3 of the
purple balloons?
(c) (4 marks) How many different choices can we make if the store has only 7 red and only 5
yellow balloons left, but 10 of each of the other colours? (This time, unlike in part (b), we
do not insist on any colours.)
2. (10 marks total) Count arrangements of the letters ANNA PAPAY A subject to the following
conditions.
(a) (3 marks) Determine the number of arrangements that contain consecutive Ps.
(b) (7 marks) Determine the number of arrangements that do not contain a consecutive Ns or
consecutive Ps.
3. (10 marks) Let s N (for N ≥ 0) be the sum of entries in the Nth row of Pascal’s triangle. (For
example, s 0 = 1.)
(a) (5 marks) Determine the number s 2022 .
(b) (5 marks) Determine the number φ(s 2022 ), where φ is Euler’s totient function.
4. (10 marks total) MATH2302 version The Round Table has twelve chairs around it in a circular
arrangement. Each chair could be either purple or gold. How many possible sequences of the
colours are there? (The chairs around the Round Table are all equal: there is no way of telling
where the sequence “starts.”)
4. (10 marks total) MATH7308 version On pages 19-20 of the course notes, you can find an
argument that leads to the formula counting the total number of circular sequences of length n
based on an alphabet of size r. What if you had an alphabet of size r = 2, but only wanted to
count sequences that had equal numbers of both letters? Modify the argument to produce such
a formula! That is, give a formula for the number of circular sequences of length n where two
letters appear n/2 times each. Then count the number of such sequences for n = 12. (For partial
credit of at most 4 points, just do the count for n = 12.)