Number of ways for A and B to have the same colour = 4

(Both colour 1)
(Both colour 2)
(Both colour 3)
(Both colour 4)


Number of ways to colour A and B without restrictions = 4 x 4 = 16

(4 possible colours for each, multiply by each other)



So, number of ways to paint them a different colour from each other

= 16 - 4
= 12


Number of ways to colour C,D and E with no restrictions

= 4 x 4 x 4
= 64


Total number of ways

= 12 x 64
= 768




Alternatively, for A and B,

There are 4 ways to choose a colour for A. This means we can only choose from 3 remaining colours for B since B cannot take that same colour we chose for A.

The order of choosing for A or B first does not matter.

Number of ways = 4 x 3 = 12

8iii)


Since there are 5 blocks and all 4 colours are used, 2 of the blocks will have the same colour. The other 3 each have unique colours.


Number of ways to choose a colour for the 2 blocks

= 4 (either colours 1,2 3 or 4)


Number of ways to assign the remaining 3 colours to the remaining 3 blocks

= 3 x 2 x 1 = 6

(3 possible blocks for 1st remaining colour 2 possible blocks for 2nd
3 possible for 3rd)


Eg.

A B C
1 2 3
1 3 2
2 3 1
2 1 3
3 1 2
3 2 1



Number of ways to choose 2 blocks for the same colour

= 5C2
= 10


Total number of ways

= 10 x 6 x 4
= 240

Alternative for 8iii)


Number ways to choose the same colour for the 2 blocks = 4


Now, we have 5 blocks where 2 have the same colour.

We have 4 colours so we assign 3 unique colours to 3 blocks and the last colour to the 2 blocks.


Eg.

Block A B C D E
Color 1 1 2 3 4

Eg.

Block A B C D E
Color 2 1 4 3 4


Number of ways to arrange the colours to assign to the 5 blocks

= 5!/2!
= 60


(1 colour is repeated so we need to divide by 2! to remove repeated cases)

Eg.

Block A B C D E
Color 1 1 2 3 4

And

Block B A C D E
Color 1 1 2 3 4

are basically indistinguishable.



Total number of ways

= 4 x 60
= 240

For part (iii) why can't you use

A B C D E
4 3 2 1 4=96?
Doesn't this ensure all of them have distinct colors too?

Are you referring to :

4 ways for A × 3 ways for B × 2 ways for C × 1 way for A × 4 ways for E?

The problem with this is, you're only making E the same colour as one of the blocks.

Eg.

Block A B C D E
Colour 1 2 3 4 1

Block A B C D E
Colour 1 2 3 4 2

Block A B C D E
Colour 1 2 3 4 3

Block A B C D E
Colour 1 2 3 4 4


This is only a portion of the total number of possible cases.


You need to consider the cases whereby :

A and B have same colour
A and C have the same colour
A and D have the same colour
B and C have same colour
B and D have the same colour
C and D have same colour