Colour Vision Deficiency is the decreased ability to see differences in colours. Someone with normal colour vision will be able to see 6 colours (Red, Orange, Yellow, Green, Blue, Violet) as 6 bands of colours. Deutan, who has colour vision deficiency, is not able to differentiate the following two pairs of colours: (I) Red and Green, (II) Blue and Violet. For 6 colours (Red, Orange, Yellow, Green, Blue, Violet) of a particular shade, when the above pairs of colours are placed next to each other, it will look like a single band of colour to Deutan. Some examples of his vision of these 6 colours are as follows.

1. Deduce the condition required to form exactly 4 bands of colour.

Normally, 6 colours placed in a line form 6 bands. For them to appear as 4 bands, there must be a reduction of 2 bands.
Since only adjacent pairs of (Red, Green) and (Blue, Violet) can merge into a single band, both pairs must be placed next to each other.

2. Calculate the number of ways to arrange the colours using the block method.

Treat the (Red, Green) pair as $1$ unit and the (Blue, Violet) pair as $1$ unit.
Including Orange and Yellow, there are $4$ units in total to arrange in a line, giving $4!$ ways.
Within their respective units, Red and Green can swap places ($2!$ ways), and Blue and Violet can swap places ($2!$ ways).
$\text{Total number of ways} = 4! \times 2! \times 2! = 24 \times 2 \times 2 = 96$

1. Find the number of arrangements where one specific pair (e.g., Red and Green) is adjacent.

Treat Red (R) and Green (G) as a single grouped unit. Together with the other 4 colours, there are 5 units to arrange.
Number of ways R and G are adjacent = $5! \times 2! = 120 \times 2 = 240$.

2. Find the number of arrangements where both problem pairs are adjacent (which results in 4 bands).

Treat R and G as one unit, and Blue (B) and Violet (V) as a second unit. Together with the 2 remaining colours, there are 4 units to arrange.
Number of ways both pairs are adjacent = $4! \times 2! \times 2! = 24 \times 2 \times 2 = 96$.

3. Calculate the number of arrangements where exactly one pair is adjacent, yielding exactly 5 bands.

To get exactly 5 bands, either ONLY R and G are adjacent, or ONLY B and V are adjacent.
Number of ways R and G are adjacent but B and V are not = $240 - 96 = 144$.
By symmetry, the number of ways B and V are adjacent but R and G are not is also $144$.
Total arrangements for exactly 5 bands = $144 + 144 = 288$.

1. Calculate the total unrestricted number of arrangements for the 6 colours.

$\text{Total unrestricted arrangements} = 6! = 720$

2. Calculate the number of arrangements where the indistinguishable pairs are adjacent.

For Deutan to see exactly 6 bands, Red & Green must not be adjacent, and Blue & Violet must not be adjacent.

Let's find the restricted arrangements:

• Number of arrangements where Red & Green are adjacent:
  Treat (Red, Green) as 1 unit $\Rightarrow 5! \times 2! = 240$
• Number of arrangements where Blue & Violet are adjacent:
  Treat (Blue, Violet) as 1 unit $\Rightarrow 5! \times 2! = 240$
• Number of arrangements where both pairs are adjacent:
  Treat both as single units $\Rightarrow 4! \times 2! \times 2! = 96$

3. Use the principle of inclusion-exclusion to find the number of ways where no indistinguishable colours are adjacent.

Using the complement method (Total $-$ ways at least one pair is adjacent):
$\text{Ways for 6 bands} = 720 - (240 + 240 - 96)$
$= 720 - 384 = 336$

(Alternatively, if relying on previous parts: $720 - [\text{Ways for 4 bands}] - [\text{Ways for 5 bands}] = 720 - 96 - 288 = 336$.)