Home
Class 12
MATHS
Suppose A(1), A(2), A(3),"………,"A(30) are...

Suppose `A_(1), A_(2), A_(3),"………,"A_(30)` are thirty sets each having 5 elements and `B_(1), B_(2), B_(3),"…………",B_(n)` are n sets each with 3 elements, let `bigcup_(i= 1)^(30)A_(i)= bigcup_(j= 1)^(n)B_(j)= S` and each elements of S belongs to exactly 10 of the `A_(i)`'s and exactly 9 of the `B_(j)'s`. Then, n is equal to

Text Solution

Verified by Experts

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
Promotional Banner

Similar Questions

Explore conceptually related problems

What does a_(1) + a_(2) + a_(3) + …..+ a_(n) represent

If a_(1),a_(2),a_(3),a_(4) and a_(5) are in AP with common difference ne 0, find the value of sum_(i=1)^(5)a_(i) " when " a_(3)=2 .

A_(1),A_(2),A_(3)...........A_(n) are arithmetic mean between two number a and b then a A_1 , A_(2) , ...............An , b becomes arithmetic sequence.

Let A_(1),A_(2),A_(3),"......."A_(m) be arithmetic means between -3 and 828 and G_(1),G_(2),G_(3),"......."G_(n) be geometric means between 1 and 2187. Product of geometric means is 3^(35) and sum of arithmetic means is 14025. The value of n is

Let A_(1),A_(2),A_(3),"......."A_(m) be arithmetic means between -3 and 828 and G_(1),G_(2),G_(3),"......."G_(n) be geometric means between 1 and 2187. Product of geometric means is 3^(35) and sum of arithmetic means is 14025. The value of m is

The y-axis divides the line segment joining A(a_(1), b_(1)) and B(a_(2), b_(2)) in the ratio . . . . .

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If A_(1),A_(2),A_(3),…,A_(n) are n points in a plane whose coordinates are (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)),…,(x_(n),y_(n)) respectively. A_(1)A_(2) is bisected in the point G_(1) : G_(1)A_(3) is divided at G_(2) in the ratio 1 : 2, G_(2)A_(4) at G_(3) in the1 : 3 and so on untill all the points are exhausted. Show that the coordinates of the final point so obtained are (x_(1)+x_(2)+.....+ x_(n))/(n) and (y_(1)+y_(2)+.....+ y_(n))/(n)

If the arithmetic mean of a_(1),a_(2),a_(3),"........"a_(n) is a and b_(1),b_(2),b_(3),"........"b_(n) have the arithmetic mean b and a_(i)+b_(i)=1 for i=1,2,3,"……."n, prove that sum_(i=1)^(n)(a_(i)-a)^(2)+sum_(i=1)^(n)a_(i)b_(i)=nab .

If a_(1), a_(2), a_(3) ,…., a_(n) are the terms of arithmatic progression then prove that (1)/(a_(1)a_(2)) + (1)/(a_(2)a_(3)) + (1)/(a_(3)a_(4)) + ….+ (1)/(a_(n-1) a_(n)) = (n-1)/(a_(1)a_(n))