Each set *X*, contains 5 elements and each set *Y*, contains 2 elements and \[\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r\] If each element of *S* belong to exactly 10 of the *X _{r}*'s and to eactly 4 of

*Y*'s, then find the value of

_{r}*n*.

#### Solution

It is given that each set *X* contains 5 elements and \[\cup^{20}_{r = 1} X_r = S\]

\[\therefore n\left( S \right) = 20 \times 5 = 100\]

But, it is given that each element of *S* belong to exactly 10 of the *X _{r}*'s.

∴ Number of distinct elements in

*S*=\[\frac{100}{10} = 10\] .....(1)

It is also given that each set *Y* contains 2 elements and \[\cup^n_{r = 1} Y_r = S\]

\[\therefore n\left( S \right) = n \times 2 = 2n\]

Also, each element of *S* belong to eactly 4 of *Y _{r}*'s.

∴ Number of distinct elements in *S* = \[\frac{2n}{4}\] .....(2)

From (1) and (2), we have

\[\frac{2n}{4} = 10\]

\[ \Rightarrow n = 20\]

Hence, the value of *n* is 20.