Key Concepts and Formulas

• Princ of Counting Element Occurrences: If $T = \bigcup_{i=1}^m S_i$ and each element $x \in T$ belongs to exactly $k$ of these sets, then the sum of the cardinalities of the individual sets is given by $\sum_{i=1}^m |S_i| = k \cdot |T|$. This formula is fundamental for problems involving overlapping sets where elements have a uniform multiplicity.
• Equality of Set Cardinality: If two different collections of sets form the same union set, the cardinality of that union set must be equal in both cases. This allows us to set up equations to solve for unknown quantities.

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Step-by-Step Solution

Step 1: Determine the cardinality of set T using the collection $X_i$.

• Understanding the given information: We are given 50 sets, $X_1, X_2, \ldots, X_{50}$. Their union is the set $T$: $\bigcup_{i=1}^{50} X_i = T$. Each set $X_i$ has 10 elements: $|X_i| = 10$. Every element in $T$ belongs to exactly 20 of these $X_i$ sets. This means the multiplicity of each element with respect to the $X_i$ sets is $k_X = 20$.

• Calculating the sum of cardinalities of $X_i$ sets: The sum of the number of elements in each $X_i$ set is: $\sum_{i=1}^{50} |X_i| = 50 \times |X_i|$ Since $|X_i| = 10$ for all $i$: $\sum_{i=1}^{50} |X_i| = 50 \times 10 = 500$ This sum represents the total count of element occurrences across all sets $X_i$.

• Applying the Principle of Counting Element Occurrences to find $|T|$: Using the formula $\sum_{i=1}^{50} |X_i| = k_X \cdot |T|$: $500 = 20 \cdot |T|$ Solving for $|T|$: $|T| = \frac{500}{20} = 25$ Thus, the total number of unique elements in set $T$ is 25.

Step 2: Set up an equation for the cardinality of set T using the collection $Y_i$.

• Understanding the given information: We are given $n$ sets, $Y_1, Y_2, \ldots, Y_n$. Their union is also the set $T$: $\bigcup_{i=1}^{n} Y_i = T$. Each set $Y_i$ has 5 elements: $|Y_i| = 5$. Every element in $T$ belongs to exactly 6 of these $Y_i$ sets. This means the multiplicity of each element with respect to the $Y_i$ sets is $k_Y = 6$.

• Calculating the sum of cardinalities of $Y_i$ sets: The sum of the number of elements in each $Y_i$ set is: $\sum_{i=1}^{n} |Y_i| = n \times |Y_i|$ Since $|Y_i| = 5$ for all $i$: $\sum_{i=1}^{n} |Y_i| = n \times 5 = 5n$ This sum represents the total count of element occurrences across all sets $Y_i$.

• Applying the Principle of Counting Element Occurrences to express $|T|$ in terms of $n$: Using the formula $\sum_{i=1}^{n} |Y_i| = k_Y \cdot |T|$: $5n = 6 \cdot |T|$ From this, we can express $|T|$ as: $|T| = \frac{5n}{6}$

Step 3: Equate the expressions for $|T|$ and solve for $n$.

• Equating the cardinalities: From Step 1, we found $|T| = 25$. From Step 2, we found $|T| = \frac{5n}{6}$. Since both expressions represent the cardinality of the same set $T$, they must be equal: $25 = \frac{5n}{6}$

• Solving for $n$: To solve for $n$, we can multiply both sides of the equation by 6: $25 \times 6 = 5n$ $150 = 5n$ Now, divide both sides by 5: $n = \frac{150}{5}$ $n = 30$ Therefore, the value of $n$ is 30.

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Common Mistakes & Tips

• Confusing Sum of Cardinalities with Union Cardinality: It's crucial to remember that $\sum |S_i|$ is not generally equal to $|\bigcup S_i|$. The former counts elements as many times as they appear in different sets, while the latter counts each unique element only once. The formula $\sum |S_i| = k \cdot |T|$ bridges this gap by incorporating the uniform multiplicity $k$.
• Misinterpreting "Exactly k": The phrase "exactly $k$" is vital. It simplifies the problem by ensuring a consistent counting factor for every element in $T$. If the condition were different (e.g., "at least $k$"), a more complex approach like the Principle of Inclusion-Exclusion would be necessary.
• Algebraic Errors: Double-check your arithmetic when solving equations, especially when dealing with fractions.

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Summary

The problem utilizes the principle that the sum of the cardinalities of a collection of sets is equal to the product of the uniform multiplicity of each element in the union and the cardinality of the union itself. By applying this principle to two different collections of sets ($X_i$ and $Y_i$) that form the same union set $T$, we derived two expressions for $|T|$. Equating these expressions allowed us to solve for the unknown number of sets, $n$.

The final answer is \boxed{30}, which corresponds to option (A).