Thinking Skills: Critical Thinking and Problem Solving

334 Answers to assignments

4    There are four ways of picking up the first

hat; then one has been removed, so there

are three ways of picking up the second

hat; or 4 × 3 = 12 ways of picking up the

first two hats. The total number of ways

they can pick up the four hats is 4 × 3 ×

2 × 1 = 24. We must subtract from this

the number where one person or more

has the right hat.

Look first at only one person having the

right hat. If A has the right hat, there

are 6 combinations of hat for B, C and

D (BCD, BDC, CBD, CDB, DBC, DCB).

Of these, only 2 have all BCD with the

wrong hats (CDB and DBC). The same

applies if B, C or D is the only person

with the right hat, making 8 in total.

Look now at two people having the

right hat: this could be AB, AC, AD, BC,

BD and CD. In each case, there is only

one way the other two could be wrong,

making 6 in total.

It is impossible for exactly three people

to have the right hats.

There is only one way all four people can

have the right hats.

This makes 8 + 6 + 1 = 15 ways of at

least one person having the right hat,

leaving 9 ways that everyone has the

wrong hat.You could try to list these.

6.2 Graphical methods of solution

1    Each row of tables contains 6 tables (6 ×

2 m = 12 m) with 1.5 m gaps at each end.

There must be a 1.5 m gap between the

wall and the first row of tables. Each

other row has an effective width of 0.8

+ 1.5 = 2.3 m. So the number of rows

that can fit in the room is the integer

below 13 5 23 .. = 5. Each row seats 6 × 6 + 2 =

38 people (the 2 are at the ends). 5 × 38 =

190, so A is correct.

2    The Venn diagram is as shown here. The

top-left circle represents even numbers,

the top-right circle multiples of 3 and

the bottom circle square numbers. Those

outside the three circles do not fit into

any of the categories.

10

20

28

38

8

26

34

214

22

32

15 21

33

3

27

(^612)
18 24
30
416
125
9
36
5
17 29
37
7
19
11
23
31 35
13
39
3 These statements may be represented as
a Carroll diagram.
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