Shirt Selection Problem Explained

This problem asks for the number of ways four individuals, Arun, Gulab, Neel, and Shweta, can each select one shirt from a set of four distinct colors: red, pink, blue, and white. The selection must adhere to specific constraints related to color preferences or dislikes.

Color Restrictions and Constraints

The problem specifies the following conditions:

• Arun dislikes the color red. This means Arun cannot choose the red shirt.
• Shweta dislikes the color white. This means Shweta cannot choose the white shirt.
• Gulab and Neel have no color restrictions; they can choose any of the available shirts.
• Each person must choose exactly one shirt, and each shirt must be chosen by exactly one person. This is a permutation problem with restrictions.

We need to find the total number of possible assignments (permutations) that satisfy these conditions.

Step-by-Step Casework Solution

One way to solve this is by considering the possible choices for Arun, who has a restriction.

Arun's possible shirt choices are Pink (P), Blue (B), or White (W), since he dislikes Red (R).

Case 1: Arun chooses Pink (A=P)

If Arun picks the pink shirt, the remaining shirts are Red, Blue, and White ({R, B, W}). These must be assigned to Gulab, Neel, and Shweta.

Shweta's restriction is that she cannot choose White (S ≠ W).

• Subcase 1.1: Shweta chooses Red (S=R). The remaining shirts are {Blue, White} for Gulab and Neel. They can be assigned in 2 ways: (G=B, N=W) or (G=W, N=B).
• Subcase 1.2: Shweta chooses Blue (S=B). The remaining shirts are {Red, White} for Gulab and Neel. They can be assigned in 2 ways: (G=R, N=W) or (G=W, N=R).

Total ways for Case 1 = 2 + 2 = 4 ways.

Case 2: Arun chooses Blue (A=B)

If Arun picks the blue shirt, the remaining shirts are Red, Pink, and White ({R, P, W}). These must be assigned to Gulab, Neel, and Shweta.

Shweta's restriction is that she cannot choose White (S ≠ W).

• Subcase 2.1: Shweta chooses Red (S=R). The remaining shirts are {Pink, White} for Gulab and Neel. They can be assigned in 2 ways: (G=P, N=W) or (G=W, N=P).
• Subcase 2.2: Shweta chooses Pink (S=P). The remaining shirts are {Red, White} for Gulab and Neel. They can be assigned in 2 ways: (G=R, N=W) or (G=W, N=R).

Total ways for Case 2 = 2 + 2 = 4 ways.

Case 3: Arun chooses White (A=W)

If Arun picks the white shirt, the remaining shirts are Red, Pink, and Blue ({R, P, B}). These must be assigned to Gulab, Neel, and Shweta.

Shweta's restriction is that she cannot choose White (S ≠ W). Since White has already been chosen by Arun, this restriction is automatically satisfied for Shweta.

Shweta can choose Red (S=R), Pink (S=P), or Blue (S=B).

• Subcase 3.1: Shweta chooses Red (S=R). Remaining shirts {Pink, Blue} for Gulab, Neel. Ways: (G=P, N=B) or (G=B, N=P). 2 ways.
• Subcase 3.2: Shweta chooses Pink (S=P). Remaining shirts {Red, Blue} for Gulab, Neel. Ways: (G=R, N=B) or (G=B, N=R). 2 ways.
• Subcase 3.3: Shweta chooses Blue (S=B). Remaining shirts {Red, Pink} for Gulab, Neel. Ways: (G=R, N=P) or (G=P, N=R). 2 ways.

Total ways for Case 3 = 2 + 2 + 2 = 6 ways.

Total Valid Ways Calculation

To find the total number of valid ways, we sum the possibilities from each case:

Total Ways = Ways(Case 1) + Ways(Case 2) + Ways(Case 3)

Total Ways = 4 + 4 + 6 = 14.

Thus, there are 14 different ways the shirts can be chosen such that no one receives a shirt of a color they dislike.

Applying the Inclusion-Exclusion Principle

Alternatively, we can use the Principle of Inclusion-Exclusion.

First, calculate the total number of ways to assign the 4 shirts (R, P, B, W) to the 4 people (A, G, N, S) without any restrictions. This is the number of permutations of 4 distinct items:

$$ \text{Total Assignments} = P(4, 4) = 4! = 4 \times 3 \times 2 \times 1 = 24 $$

Now, let $P_A$ be the property that Arun chooses the Red shirt, and $P_S$ be the property that Shweta chooses the White shirt.

Calculate the number of assignments where Arun chooses Red ($N(P_A)$):

If Arun is assigned Red, the remaining 3 shirts (P, B, W) can be assigned to the other 3 people (G, N, S) in $3!$ ways.

$$ N(P_A) = 3! = 3 \times 2 \times 1 = 6 $$

Calculate the number of assignments where Shweta chooses White ($N(P_S)$):

If Shweta is assigned White, the remaining 3 shirts (R, P, B) can be assigned to the other 3 people (A, G, N) in $3!$ ways.

$$ N(P_S) = 3! = 3 \times 2 \times 1 = 6 $$

Calculate the number of assignments where Arun chooses Red AND Shweta chooses White ($N(P_A \cap P_S)$):

If Arun is assigned Red and Shweta is assigned White, the remaining 2 shirts (P, B) can be assigned to the other 2 people (G, N) in $2!$ ways.

$$ N(P_A \cap P_S) = 2! = 2 \times 1 = 2 $$

Using the Principle of Inclusion-Exclusion, the number of assignments where at least one restriction is violated is:

$$ N(P_A \cup P_S) = N(P_A) + N(P_S) - N(P_A \cap P_S) $$ $$ N(P_A \cup P_S) = 6 + 6 - 2 = 10 $$

The number of valid assignments where neither restriction occurs is the total assignments minus the assignments where at least one restriction occurs:

$$ \text{Valid Ways} = \text{Total Assignments} - N(P_A \cup P_S) $$ $$ \text{Valid Ways} = 24 - 10 = 14 $$

Both methods confirm that there are 14 valid ways for the shirts to be chosen.