Combinatorics Worksheet 1
All SMC, BMO and Mentoring problems are © UKMT (www.ukmt.org.uk)
1. How many ways of arranging the letters in BANANA?
2. How many ways are there of writing a 5 letter word:
a. where no two letters are the same?
b. where no two adjacent letters are the same?
3. [SMC] In how many different ways can I circle letters in the grid shown so that there is
exactly one circled letter in each row and exactly one circled letter in each column?
A
D
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
15
100
B
E
24
120
C
60
4. The well-known ‘Birthday Paradox’ asks the following question: “How many people do you
need in a room such that there’s about a 50% chance that two people share the same
birthday?”. By using the example in the slides (under ‘Probabilistic vs Combinatoric
approaches) as a guide, show that when there are 23 people, the probability is roughly a
half, using:
a. A probabilistic approach.
b. A combinatoric approach.
(Hint: When determining the probability of ‘at least one thing occurring’, it’s much easier to
find the probability of the event never occurring and subtracting from 1; in this case no one
sharing a birthday)
5. An examiner invigilates an exam in which 1500 students are applying for 150 places at a
school. The examiner is responsible for invigilating a room of 25 students. What is the
chance that all of them get a place? Use:
a. A probabilistic approach.
b. A combinatoric approach.
6. I have two identical packs of cards. I take each pack and shuffle it separately, placing them
both face down in the table. I then proceed to play a game of snap with myself (being a
mathematician I have no friends). I compare the top card from each pile, and then compare
the second card from each pile, and so on. What is the probability that I continue in this way
all the way down the piles, and never find an exact match in the 52 pairs of cards?
www.drfrostmaths.com/rzc