Out of a class of 100 students, 25 play at least cricket and football, 15 play at least cricket and hockey, 12 play at least football and hockey and 10 play all the three sports. The number of students playing cricket, football and hockey are 50, 37 and 22, respectively. The number of students who do NOT play any of the three sports is

Correct — A, 33. The three-set inclusion-exclusion rule says the number playing at least one game is the sum of the three single totals, minus the three pairwise totals, plus the triple total: 50 + 37 + 22 − 25 − 15 − 12 + 10. The singles add to 109, the pairs subtract 52 to leave 57, and adding back the 10 who play all three gives 67. Out of the class of 100, that leaves 100 − 67 = 33 students playing none of the three sports. The phrase 'at least' in each pairwise figure is what makes the rule apply straight off, since it means the 25, 15 and 12 already include the 10 triple players.

• (b)23 — 23 comes from adding the triple-count twice, or from subtracting the pairwise totals without restoring the 10 who play all three.
• (c)27 — 27 would need 73 players in the union. No arrangement of the given figures produces that total.
• (d)30 — 30 is a round guess. It would need the union to be exactly 70, which is 3 more than the 67 the figures give.

For three overlapping groups, the size of the union is |A| + |B| + |C| − |A∩B| − |B∩C| − |C∩A| + |A∩B∩C|. Each element of a pairwise overlap has been counted twice in the singles, so it is removed once; each element of the triple overlap has been counted three times and removed three times, so it is added back once. Whatever is outside the union is the complement, found by subtracting from the total population.

Everything turns on how the pairwise numbers were reported. 'At least cricket and football' means every student who plays both, whether or not they also play hockey, so the 25 already contains the 10 triple players and the formula can be applied unchanged. Had the paper said 'exactly cricket and football', those 10 would have had to be added into each pair first, and the union would have come to a different figure. Reading that phrase correctly is the whole discrimination in the item.

• The union of three sets is the singles minus the pairs plus the triple.
• Here: 50 + 37 + 22 = 109; minus 25 + 15 + 12 = 52; plus 10 gives 67.
• Students playing none = 100 − 67 = 33.
• 'At least A and B' counts everyone in the intersection, including those in all three groups.
• Only 50 − 25 − 15 + 10 = 20 students play cricket alone, by the same rule applied to one set.

• Reading 'at least cricket and football' as 'only cricket and football'.
• Forgetting to add the triple overlap back after subtracting the pairs.
• Answering with the size of the union instead of the number outside it.

A set-theory item whose numbers are straightforward and whose difficulty lies entirely in one phrase of the stem.

No directly related past PYQ was found.

• practice — not a real PYQ

  In a group of 60 people, 30 read Hindi papers, 25 read English papers and 10 read both. How many read neither?

  • (a)10
  • (b)15
  • (c)20
  • (d)25

  Answer(b) 15 — the union is 30 + 25 − 10 = 45, so 60 − 45 = 15 read neither.

• practice — not a real PYQ

  In the class described above, how many students play cricket alone?

  • (a)10
  • (b)15
  • (c)20
  • (d)25

  Answer(c) 20 — 50 − 25 − 15 + 10 = 20, restoring the triple players once after removing both pairs.