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 gets a shirt in a colour he or she dislikes? A. 21 B. 18 C. 16 D. 14

GATE 2017 · General Aptitude · Combinatorics · medium

Answer: 14 ways (option D)

1. Unrestricted total: 4 people, 4 distinct shirts: total = 4! = 24.
2. Subtract: Arun gets red (|A|): Fix Arun with red; remaining 3 people arrange 3 shirts freely: 3! = 6.
3. Subtract: Shweta gets white (|B|): Fix Shweta with white; remaining 3 arrange 3 shirts freely: 3! = 6.
4. Add back double-counted: both happen (|A intersect B|): Fix Arun with red AND Shweta with white; remaining 2 (Gulab, Neel) arrange 2 shirts: 2! = 2.
5. Apply inclusion-exclusion: Invalid = |A| + |B| - |A intersect B| = 6 + 6 - 2 = 10. Valid = 24 - 10 = 14.