at break,in a school123 students go to canteen which sells cakes,ice cream and buns.

42 students buy ice-cream, 36 buy bunsand 10 buy only cakes, 15 students buy ice-cream and buns, 10 buy ice-cream and cakes 4 buy cakes and buns

but not ice creamand 11 buy ice cream and buns but no cakes draw venn diagram to illustratethe above information and find

1)how many students buy nothing at all

2)how many students buy at least  two items.

3)how many students buy all three items.

please can u ans this..

Let U denotes the universal set.

Let us denote the following sets as:

A: students who ate cake only

B: students who ate ice - cream only

C: students who ate buns only

D: students who ate ice-cream and cake but not buns

E: students who ate cakes and buns but not ice-cream

F: students who ate ice-cream and buns but not cake

G: students who ate cakes, buns and ice-cream

Then according to the given information we have:

n(D) + n(B) + n(G) + n(F) = 42

n(E) + n(G) + n(F) + n(C) = 36

n(A) = 10

n(G) + n(F) = 15

n(D) + n(G) = 10

n(E) = 4

n(F) = 11

On solving above, we get n(G) = 4, n(D) = 6, n(C) = 17, n(B) = 21.

(1)

Number of students who buy cakes, buns and ice-cream

 = n(A) + n(B) + n(C) + n(D) + n(E) + n(F) + n(G)

= 10 + 21 + 17 + 6 + 4 + 11 + 4

= 73

∴ 

Number of students who buy nothing at all = n(U) – 73 = 123 – 73 = 50

(2) Number of students who buy at least 2 items  = n(D) + n(E) + n(G) + n(F)

                                                                              = 6 + 4 + 4 + 11

                                                                              = 25

(3) Number of students who buy all 3 items = n(G) = 4