GATE CS 2017 Set 1 QGA-9 — General Aptitude

We need to find the number of ways to distribute four distinct shirts—Red ($R$), Pink ($P$), Blue ($B$), and White ($W$)—to four people—Arun ($A$), Gulab ($G$), Neel ($N$), and Shweta ($S$)—such that:

• Arun ($A$) does not get Red ($R$).
• Shweta ($S$) does not get White ($W$).
• Gulab ($G$) and Neel ($N$) can get any color.

We can solve this using the Principle of Inclusion-Exclusion (PIE):

Let $U$ be the set of all possible distributions of the 4 shirts to the 4 people without any restrictions.
$$\text{Total unrestricted ways } |U| = 4! = 24$$

Let $A_R$ be the set of distributions where Arun gets the Red shirt:
If Arun gets $R$, the remaining 3 shirts can be distributed among $G, N, S$ in $3! = 6$ ways.
$$|A_R| = 6$$

Let $S_W$ be the set of distributions where Shweta gets the White shirt:
If Shweta gets $W$, the remaining 3 shirts can be distributed among $A, G, N$ in $3! = 6$ ways.
$$|S_W| = 6$$

Let $A_R \cap S_W$ be the set of distributions where Arun gets Red and Shweta gets White:
If Arun gets $R$ and Shweta gets $W$, the remaining 2 shirts can be distributed among $G$ and $N$ in $2! = 2$ ways.
$$|A_R \cap S_W| = 2$$

Using the Principle of Inclusion-Exclusion, the number of valid ways is: $$\text{Valid Ways} = |U| - (|A_R| + |S_W| - |A_R \cap S_W|)$$ $$\text{Valid Ways} = 24 - (6 + 6 - 2) = 24 - 10 = 14$$

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