Each set X, contains 5 elements and each set Y, contains 2 elements and $⋃_{r = 2}^{20} X_{r} = S = ⋃_{r = 1}^{n} Y_{r}$ . If each element of S belongs to exactly 10 of the $X_{r}^{^{'} s}$ and to exactly 4 of $Y_{r}^{^{'} s}$ then find the value of n.

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Solution

If elements are not repeated, then number of elements in $X_{1} \cup X_{2} \cup X_{3} . . . . . \cup X_{20}$ is $20 \times 5$

But each elemets is used 10 times , so

$S = \frac{20 \times 5}{10} = 10$

If elements in $Y_{1}, Y_{2}, . . . . . Y_{n}$ are not repeated, then total number of elements is 2n but each element is repeated 4 times , so

$S = \frac{2 n}{4} \Rightarrow 10 = \frac{2 n}{4}$

n = 20

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\cup_{r = 1}^{20} X_{r} = S = \cup_{r = 1}^{n} Y_{r}

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Q. Each set X, contains 5 elements and each set Y, contains 2 elements and

\cup_{r = 1}^{20} X_{r} = S = \cup_{r = 1}^{n} Y_{r}

. If each element of S belong to exactly 10 of the X_r's and to eactly 4 of Y_r's, then find the value of n.

Suppose $A_{1}, A_{2}, . . ., A_{30}$ are thirty sets each having 5 elements and $B_{1}, B_{2}, . . ., B_{n}$ are n sets each having 3 elements. Let each element of S belongs to exactly 10 of $A_{i}^{'}$ s and exactly 9 of $B_{i}^{'}$ s. Then, find the value of n.