Question

competition medals, 37 dance, 13 drama and 19 music. 46 total recipients. 5 got 3 different medals. How many received medals in exactly two of these categories?

Answer 1

We know that the total number of recipeints is 46

A, B and C are the number people who recieved medals in exactly two of these categories

we can write it as

(37-A-B)+(19-B-C)+(13-A-C)+(A+B+C)+5 = 46

this gives

(37+13+19+5) -2A-2B-2C +A+B+C = 46

74 - A-B-C = 46

or

74 - (A+B+C) = 46

Adding (A+B+C) on both sides, we get

74 - (A+B+C)+(A+B+C) = 46+(A+B+C)

this gives

74 = 46+ (A+B+C)

subtracting 46 on each side, we get

(A+B+C) = 74- 46 = 28

so, there are 28 people who recieved medals in exactly two of these categories