Question 64

There are 4 people: Arun (A), Gulab (G), Neel (N), and Shweta (S).
There are 4 shirts: Red (R), Pink (P), Blue (B), and White (W).

The total number of bijective assignments of the 4 distinct shirts to the 4 distinct people without any restriction is:

Identify the disallowed conditions:

• Condition $E_A$: Arun gets the Red shirt.
• Condition $E_S$: Shweta gets the White shirt.

We seek the number of assignments satisfying neither condition: $|E_A^c \cap E_S^c|$.

By the Principle of Inclusion-Exclusion:

1. Number of assignments where Arun gets Red ($|E_A|$):
   Arun's shirt is fixed as Red. The remaining 3 shirts are distributed among Gulab, Neel, and Shweta in $3! = 6$ ways.

2. Number of assignments where Shweta gets White ($|E_S|$):
   Shweta's shirt is fixed as White. The remaining 3 shirts are distributed among Arun, Gulab, and Neel in $3! = 6$ ways.

3. Number of assignments where both occur ($|E_A \cap E_S|$):
   Arun gets Red and Shweta gets White. The remaining 2 shirts (Pink, Blue) are distributed between Gulab and Neel in $2! = 2$ ways.

Substituting these values:

Therefore, there are 14 different ways to choose the shirts.