Discrete math question A professor gave his 40 students a te

Discrete math question

A professor gave his 40 students a test with three questions. Every student answered at least one question. Ten student did not answer the first question, question. If 18 students answered all three questions, how many answered exactly one question?

Solution

as given let assume that,

A=number of students answered just the first question and nothing else

B=number of students answered just the second question and nothing else

C=number of students answered just the third question and nothing else

D = number of students answered all three = 18

E = number of students just answered 1 and 2 but not 3

F = number of students just answered 1 and 3 but not 2

G = number of students just answered 2 and 3 but not 1

Ten student did not answer the first question means 10 students answered second and third question or just second question or just thirs question so

we can say that B + C + G = 10

14 student did not answer the second question means 14 students answered first and third question or just first question or just third question so

we can say that A + C + F = 14

12 student did not answer the third question means 12 students answered first and second question or just first question or just second question so

we can say that A + B + E = 12

adding this three equation we have,

B + C + G + A + C + F + A + B + E = 10+14+12

2A + 2B + 2C + G + F + E = 36 --------------1)

as 40 students apply for a test we can say that,

A + B + C + D + E + F + G = 40 -----------------2)

subtract 2) from 1) we have,

2A + 2B + 2C + G + F + E - A - B - C - D - E - F - G = 36 - 40

we have D = 18 so,

A + B + C - 18 = -4

A + B + C = 14

But A ,B , C is the number of students answered only one question.

so we can say that 14 students has answered only 1 question