एक ग्राम में आधे ग्रामवासियों के अपने घर हैं । 1/5 ग्रामवासी धान की खेती करते हैं । 1/3 ग्रामवासी साक्षर हैं । 4/5 ग्रामवासियों की आयु 25 वर्ष से कम है । निम्नलिखित कथनों में से कौन-सा निश्चित रूप से सही है ?
सही उत्तर: (b) 25 वर्ष से कम आयु वाले कुछ ग्रामवासी साक्षर हैं ।
मान लीजिए गाँव की कुल जनसंख्या N है। घर वाले = N/2, धान खेती = N/5, साक्षर = N/3, 25 वर्ष से कम आयु वाले = 4N/5। निरक्षर व्यक्तियों की संख्या = N - N/3 = 2N/3। चूँकि 25 वर्ष से कम आयु वाले (4N/5) और निरक्षर (2N/3) का योग कुल जनसंख्या N से अधिक है, समुच्चय सिद्धांत (inclusion-exclusion) के अनुसार कम से कम कुछ व्यक्ति ऐसे अवश्य होंगे जो 25 वर्ष से कम आयु के भी हैं और साक्षर भी हैं। अतः विकल्प (b) निश्चित रूप से सही है।
In English (Question & Model Answer)
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct?
Correct Option: (b) Some villagers under 25 years of age are literate.
Let the total population of the village be N. Given: 1. Half of the villagers have their own houses (N/2). 2. One-fifth cultivate paddy (N/5). 3. One-third are literate (N/3). 4. Four-fifth are under 25 years of age (4N/5). Let us analyze the overlap between villagers under 25 years of age and literate villagers. Total villagers under 25 = 4N/5. Total illiterate villagers = N - (N/3) = 2N/3. Maximum possible illiterate villagers who are under 25 cannot exceed the total number of illiterates, which is 2N/3. Since 4N/5 + 2N/3 = (12N + 10N)/15 = 22N/15 > N, by the Principle of Inclusion-Exclusion, the overlap (villagers who are both under 25 and illiterate) is at least (22N/15) - N = 7N/15 > 0. This proves that there must be some villagers who are under 25 and literate. Therefore, statement (b) 'Some villagers under 25 years of age are literate' is certainly correct. Other options cannot be logically deduced with certainty.