Combinatorics Levelled Activity

Instructions: You may work in pairs. Start on the Level 1 problems provided, recording your answer on this sheet. Answers to Level 1 questions are at the front of the class. Once you’ve checked your answers, go to your teacher and ask for Level 2.

You will need to ask the teacher to check your answers to Level 2 and beyond.

Level 1

Question / Answer
1
2
3
4
5
6
7
8

Level 2

Question / Answer
1
2
3
4
5

Level 3

Question / Answer
1
2
3
4
5
6
7

Combinatorics Levelled Activity

Instructions: You may work in pairs. Start on the Level 1 problems provided, recording your answer on this sheet. Answers to Level 1 questions are at the front of the class. Once you’ve checked your answers, go to your teacher and ask for Level 2.

You will need to ask the teacher to check your answers to Level 2 and beyond.

Level 1

Question / Answer
1
2
3
4
5
6
7
8

Level 2

Question / Answer
1
2
3
4
5

Level 3

Question / Answer
1
2
3
4
5
6
7

Level 1

Note that for larger numbers you are permitted to leave your answer in terms of powers and factorials.

  1. How many possible 3 letters ‘words’ are there using letters from the English alphabet.
  2. I take 4 items from 7 and put them in a line. How many possibilities are there?
  3. From 9 people I choose 4 people to form a team. How many possible teams are there?
  4. How many squares of any size are in this diagram?
  5. How many ways can I arrange 6 books on a shelf?
  6. How many ways can you arrange the letters of the word CATTLE?
  7. [JMC 2000 Q11] The digits of this year, 2000 A.D., add up to 2. In how many other years since 1 A.D. has this happened?
  8. [JMC 2007 Q7] The equilateral triangle is fixed in position. Two of the four small triangles are to be painted black and the other two are to be painted white. In how many different ways can this be done?

Level 2

  1. How many possible combinations of outcomes are there from the throw of three dice and five coins?
  2. [JMO 2006 A4] An equilateral triangle is drawn on a sheet of white card and divided into three identical regions as shown. Then each region is painted red or yellow or blue. More than one region may be painted in the same colour. How many different painted triangles can be made in this way? (Rotating a triangle does not make it different.)
  3. [JMO 2008 A4] How many three-digit numbers have the product of their digits equal to 6?
  4. [TMC Regional 2012 Q9] In how many ways can you split your team of 4 people into two separate teams, so that there is at least one person in each team?
  5. [Kangaroo Grey 2009 Q16] How many ten-digit numbers are there which contain only the digits 1, 2 or 3, and in which any pair of adjacent digits differs by 1?
    A 16B 32C 64D 80E 100

Level 3

  1. [Cayley 2008 Q1] How many four-digit multiples of 9 consist of four different odd digits?
  2. [IMC 2010 Q24] A new taxi firm needs a memorable phone number. They want a number which has a maximum of two different digits. Their phone number must start with the digit 3 and be six digits long. How many such numbers are possible?
    A 288B 280C 279D 226E 225
  3. [SMC 2005 Q16] A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are 4 substitutes: 1 goalkeeper, 1 defender, 1 midfielders and 1 forward. A substitute may only replace a player of the same category, e.g. midfielder for midfielders. Given that a maximum of 3 substitutes may be used and that there are still 11 players on the pitch at the end, how many different teams could finish the game?
    A 110B 118C 121D 125E 132
  4. [JMO 2004 A10] The Famous Five have been given 20 sweets as a reward for solving a tricky crime. They have agreed that the oldest of them must receive more than the next oldest, who must receive more than the next oldest, and so on. Assuming that each of the five gets at least one sweet, in how many different ways can they share the sweets?
  5. [JMO 2002 B1] A number like 4679 is called an ascending number because each digit in the number is larger than the preceding one.
    (i) How many ascending numbers are there between 1000 and 2000?
    (ii) How many ascending numbers are there between 1000 and 10000?
  6. [Kangaroo Pink 2007 Q24] At a party, five girls give each other gifts in such a way that everybody gives one gift and everybody receives one (though of course nobody receives their own gift). How many possible ways are there for this to happen?
    A 5B 10C 44D 50E 120
  7. [SMC 2001 Q21] A postman’s sack contains five letters, one each for the five houses in Cayley Close. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?
    A 5B 10C 20D 30E 60