Arun, Gulab, Neel and Shweta must choose one shirt each from a pile of four shirts coloured red, pink, blue and white respectively. Arun dislikes the colour red and Shweta dislikes the colour white. Gulab and Neel like all the colours. In how many different ways can they choose the shirts so that no one has a shirt with a colour he or she dislikes?

We need to find the number of one-to-one assignments of $4$ shirts $\{\text{Red, Pink, Blue, White}\}$ to $4$ people $\{\text{Arun (A), Gulab (G), Neel (N), Shweta (S)}\}$ such that:

• Arun does not get Red ($A \neq R$)

• Shweta does not get White ($S \neq W$)

We can use the Principle of Inclusion-Exclusion:

1. Total unrestricted assignments:

   $$N = 4! = 24$$

2. Number of ways Arun gets Red ($E_A$):

   $$|E_A| = 3! = 6$$

3. Number of ways Shweta gets White ($E_S$):

   $$|E_S| = 3! = 6$$

4. Number of ways both Arun gets Red AND Shweta gets White ($E_A \cap E_S$):

   $$|E_A \cap E_S| = 2! = 2$$

5. Number of valid ways:

   $$\text{Valid Ways} = N - (|E_A| + |E_S| - |E_A \cap E_S|) = 24 - (6 + 6 - 2) = 24 - 10 = 14$$

Therefore, there are 14 valid ways.