Suppose `A_1,A_2….. A_(30)` are thirty sets each having 5 elements and `B_1B_2…..B_n` are n sets each having 3 elements ,Let `overset(30)underset(i=1)bigcupA_1=overset(n)underset(j=1)bigcupB_j=s`
and each element of S belongs to exactly 10 of the `A_1` and exactly 9 of the value of n.

Explanation

Given, A's are thirty sets with five elements each, so
`underset(i=1)overset(30)(Sigma)n(A_(i))=5xx30=150" ... (i)"`
If the m distinct elements in S and each element of S belongs to exactly 10 of the `A_(i)'s`, so we have
`underset(i=1)overset(30)(Sigma)n(A_(i))=10m" ... (ii)"`
`therefore` From Eqs. (i) and (ii), we get 10m = 150
`therefore m = 15 " ... (iii)"`
Similarly, `underset(j=1)overset(n)(Sigma)n(B_(j))=3n and underset(j=1)overset(n)(Sigma)n(B_(j))=9m`
`therefore 3n=9mimpliesn=(9m)/(3)=3m`
`=3xx15=45` [from Eq. (iii)]
Hence, n = 45