In a group of 950 persons, 750 can speak Hindi and 460 can speak English.
Question:

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:

(i) how many can speak both Hindi and English:

(ii) how many can speak Hindi only;

(iii) how many can speak English only.

Solution:

Let A & B denote the sets of the persons who like Hindi & English, respectively.

Given:

$n(A)=750$

$n(B)=460$

$n(A \cup B)=950$

(i) We know:

$n(A \cup B)=n(A)+n(B)-n(A \cap B)$

$\Rightarrow 950=750+460-n(A \cap B)$

$\Rightarrow n(A \cap B)=260$

Thus, 260 persons can speak both Hindi and English.

(ii) $n(A-B)=n(A)-n(A \cap B)$

$n(A-B)=750-260=490$

Thus, 490 persons can speak only Hindi.

(iii) $n(B-A)=n(B)-n(A \cap B)$

$\Rightarrow n(B-A)=460-260$

$=200$

Thus, 200 persons can speak only English.

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