B%29 Combinatorics - Worksheet 1

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