Question

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows: French = 17, English = 13, Sanskrit = 15, French and English = 09, English and Sanskrit = 4,French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study (i) French only (ii) English only (iii) Sanskrit only (iv) English and Sanskrit (v) French and Sanskrit but not English (vi) French and English but not Sanskrit (vii) at least one of the three languages (viii) none of the three languages.(without using venn diagram)

Answer 1

Answer:  The answers are given below.

Step-by-step explanation:  Given that there are total 50 students in a group.

Let 'F', 'E' and 'S' represents the set of students who study French, English and Sanskrit respectively.

So, from the given information, we have

$n(F)=17,~~n(E)=13,~~n(S)=15,~~n(F\cap E)=9,~~n(E\cap S)=4,~~n(F\cap S)=5,\\n(F\cap E\cap S)=3.$

(i) The number of students who study French only is given by

$n(F)-n(F\cap E)-n(F\cap S)+n(F\cap E\cap S)\\\\=17-9-5+3\\\\=20-14\\\\=6.$

(ii) The number of students who study English only is given by

$n(E)-n(F\cap E)-n(E\cap S)+n(F\cap E\cap S)\\\\=13-9-4+3\\\\=16-13\\\\=3.$

(iii) The number of students who study Sanskrit only is given by

$n(S)-n(F\cap S)-n(E\cap S)+n(F\cap E\cap S)\\\\=15-5-4+3\\\\=18-9\\\\=9.$

(iv) The number of students who study English and Sanskrit but not French is given by

$n(E\cap S)-n(F\cap E\cap S)\\\\=4-3\\\\=1.$

(v) The number of students who study French and Sanskrit but not English is given by

$n(F\cap S)-n(F\cap E\cap S)\\\\=5-3\\\\=2.$

(vi) The number of students who study French and English but not Sanskrit   is given by

$n(F\cap E)-n(F\cap E\cap S)\\\\=9-3\\\\=6.$

(vii) The number of students who study at least one of the three languages is given by

$n(F\cup E\cup S)=n(F)+n(E)+n(S)-n(F\cap E)-n(E\cap S)-n(F\cap S)+n(F\cap E\cap S)\\\\\Rightarrow n(F\cup E\cup S)=17+13+15-9-4-5+3=48-18=30.$

(viii) The number of students who study none of the languages is given by

$\textup{total number of students}-n(F\cup E\cup S)=50-30=20.$

Answer 2

Step-by-step explanation:

answer above....

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