In a village of 800 persons, each person plays at least one of the three sports: Cricket, Football, and Hockey. The percentages of persons playing each sport are as follows: Cricket - 43%, Football - 52.5%, and Hockey - 55%. Additionally, 96 persons play both Cricket and Football but not Hockey, 100 persons play both Cricket and Hockey but not Football, 88 persons play both Hockey and Football but not Cricket, and 60 persons play all three sports.

How many persons play neither Football nor Cricket?

Question

In a village of 800 persons, each person plays at least one of the three sports: Cricket, Football, and Hockey. The percentages of persons playing each sport are as follows: Cricket - 43%, Football - 52.5%, and Hockey - 55%. Additionally, 96 persons play both Cricket and Football but not Hockey, 100 persons play both Cricket and Hockey but not Football, 88 persons play both Hockey and Football but not Cricket, and 60 persons play all three sports.

How many persons play neither Football nor Cricket?

Accepted Answer

**Solution:**

1. Total persons = 800

2. Number of persons playing each sport:
   - Cricket (C) = 43% of 800 = $0.43 	imes 800 = 344$
   - Football (F) = 52.5% of 800 = $0.525 	imes 800 = 420$
   - Hockey (H) = 55% of 800 = $0.55 	imes 800 = 440$

4. Let us find the number of persons playing only Cricket, only Football, and only Hockey.

Using the formula for the number of persons playing Cricket:
   $$|C| = |C 	ext{ only}| + |C \cap F \cap H^c| + |C \cap H \cap F^c| + |C \cap F \cap H|$$

So,
   $$|C 	ext{ only}| = |C| - (|C \cap F \cap H^c| + |C \cap H \cap F^c| + |C \cap F \cap H|)$$
   $$= 344 - (96 + 100 + 60) = 344 - 256 = 88$$

5. Similarly, for Football:
   $$|F| = |F 	ext{ only}| + |C \cap F \cap H^c| + |H \cap F \cap C^c| + |C \cap F \cap H|$$

$$|F 	ext{ only}| = 420 - (96 + 88 + 60) = 420 - 244 = 176$$

6. For Hockey:
   $$|H| = |H 	ext{ only}| + |C \cap H \cap F^c| + |H \cap F \cap C^c| + |C \cap F \cap H|$$

$$|H 	ext{ only}| = 440 - (100 + 88 + 60) = 440 - 248 = 192$$

7. The question asks for the number of persons who play neither Football nor Cricket. Since every person plays at least one sport, those who play neither Football nor Cricket must be those who play only Hockey.

8. Therefore, the number of persons who play neither Football nor Cricket = $|H 	ext{ only}| = 192$

**Note:** The options given include 192, but the correct answer based on calculation is 192. However, the question asks for the number who play neither Football nor Cricket, which is the number playing only Hockey.

**Answer:** 192

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