Suppose are 30 sets (each with five elements) and are n sets (each with three elements) such that and each element of belong to exactly ten of the 's and exactly 9 of the 's. Then is equal to

45 Explanation: Step 1: Calculate the total number of elements in sets A (with repetitions) We have sets, and each set contains elements. Total count of elements in all sets (counting every time an element appears) is: Step 2: Find the number of unique elements in set S using A We are told that every element in the union set belongs to exactly of the sets. Let be the number of distinct elements in . The total count from Step 1 can also be expressed as: So, the set contains 15 unique elements . Step 3: Calculate the total number of elements in sets B (with repetitions) We have sets, and each set contains elements. Total count of elements in all sets is: Step 4: Find n using the unique elements of S We are told that every element in belongs to exactly of the sets. Since we know there are unique elements in

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NIMCET 2021 — Mathematics PYQ

NIMCET | Mathematics | 2021

Suppose $A_{1}, A_{2}, ..., A_{30}$ are 30 sets (each with five elements) and $B_{1}, B_{2}, B_{3}, ..., B_{n}$ are n sets (each with three elements) such that $⋃_{i = 1}^{30} A_{i} = ⋃_{j = 1}^{n} B_{j} = S$ and each element of $S$ belong to exactly ten of the $A_{i}$ 's and exactly 9 of the $B_{j}$ 's. Then $n$ is equal to

NIMCET 2021 — Mathematics PYQ

Choose the correct answer:

Explanation

Step 1: Calculate the total number of elements in sets A (with repetitions)

Step 2: Find the number of unique elements in set S using A

Step 3: Calculate the total number of elements in sets B (with repetitions)

Step 4: Find n using the unique elements of S

Step 5: Solving for n

Final Answer