Question

in a survey of 100 students, the number of students studying the various languages were found to be as follows: English only 18, English but not Hindi 23, English and French 8, English 26, French 48, French and Hindi 8 and No language 24. How many students were studying Hindi and How many students were studying English and Hindi?

Answer 1

Is it multiple choice?

Answer 2

Let \(E\) = English
\(H\) = Hindi
\(F \) = French

\(n(E \cap \overline{F} \cap \overline{H})=18\)
\(n(E \cap \overline{H})=23\)
\(n(E \cap F)= 8\)
\(n(E)=26\)
\(n(F)=48\)
\(n(F\cap H)=8\)
\(n(\overline{E} \cap \overline{F} \cap\overline{H} )=24\)

\(n(E \cap H) = ?\)

So:
\(n(\overline{E} \cap \overline{F} \cap\overline{H} )=n(\overline{E \cup F \cup H})\) by DeMorgan's Law.


\(n(\overline{E \cup F \cup H})=1-n(E \cup F \cup H)\) by complements.

\[ n(E \cup F \cup H)=n(E)+n(F)+n(H)-n(E \cap F)-n(E \cap H) \\~~~~~~~~~~~~~~~~~~~~~~~~~~-n(F \cap H) + n(E \cap F \cap H)\\
~ \\100-24 =1-[26+48+n(H)-8-n(E \cap H)-8+n(E \cap F \cap H)]\]

Notice though that \(n(E)=n(E \cap H)+ n(E \cap \overline {H})\) giving \(n(E \cap H)=3+0=3\)

Answer 3

Woah.. I am so sorry// I don't understand this at all .

Answer 4

oops that should say 100 - 76 on the before last line