SOLUTION: A survey of 100 students at New England College showed the following: 45 take English. 47 take history. 37 take language. 13 take English and history. 17 take English and

Let E = the set of students learning English, and let n(E)= 45 be the number of elements in this set.

Let H = the set of students learning History, and let n(H)= 47 be the number of elements in this set.

Let L = the set of students learning Language, and let n(L)= 37 be the number of elements in this set.

Let EH = the intersection of E and H, and let n(EH)= 13 be the number  of elements in this set.

Let EL = the intersection of E and L, and let n(EL)= 17 be the number  of elements in this set.

Let HL = the intersection of H and L, and let n(EL)= 19 be the number  of elements in this set.

Let EHL = the intersection of E, H and L, and let n(EHL)= 7 be the number  of elements in this set.

(a) Take history but neither of the other two?
 
        = n(H) - n(EH) - n(HL) + n(EHL) = 47 - 13 - 19 + 7 = 22.


(b) Take English and history but not language?
 
        = n(EH) - n(EHL) = 13 - 7 = 6.


(c) Take none of the three?
 
        = 100 - [n(E) + n(H) + n(L) - N(EH) - n(HL) - n(EL) + n(EHL)] = calculate it on your own: simply substitute data

                (What you see in BRACKETS is the number of those who takes at least one subject)


(d) Take just one of the three?
 
        = [n(E)-n(EH)-n(EL)+n(EHL)] + [n(L)-n(EL)-n(HL)+n(EHL)] + [n(H)-n(EH)-n(HL)+n(EHL)]] = calculate it on your own: simply substitute data


(e) Take exactly two of the three?
 
        = [n(EH) - n(EHL)] + [n(HL) - n(EHL)] + [n(EL) - n(EHL)] = calculate it on your own: simply substitute data


(f) Do not take language?
 
        = 100 - n(L) = 100 - 37 = 63.