Question

Q.33 A college awarded 38 medals in football, 15 in basketball and 20 in cricket.If these medals

went to a total of 58 men and only three men got medals in all the three sports,How many

received medals in exactly two of three sports?​

Answer 1

Step-by-step explanation:

Let, F=football, B=basketball and C= cricket.

Then n(F)=38, n(B)=15 and n(C)=20, n(F∪B∪C)=58, n(F∩B∩C)=3.

Since, n(F∪B∪C)= [n(F)+n(B)+n(C)]−[n(F∩B)+n(B∩C)+n(C∩F)]+n(F∩B∩C).

or, 58=(38+15+20)−[n(F∩B)+n(B∩C)+n(C∩F)]+3

or, [n(F∩B)+n(B∩C)+n(C∩F)]=76−58=18

Let, a be the no. of men who play only football and basketball,

b be the no. of men who play only football and cricket,

and c be the no. of men who play only cricket and basketball.

From, venn-diagram,

n(F∩B)=a+3

n(B∩C)=b+3

n(C∩F)=c+3

∴[n(F∩B)+n(B∩C)+n(C∩F)]=a+b+c+9

or, 18=a+b+c+9

a+b+c=9

9 men received medals in exactly two of the three sports.