In a group of 400 boys, each boy plays atleast one of the games cricket, hockey and football. 185 play cricket, 165 play hockey and 160 play football. 40 boys play only cricket and football. 20 boys play only hockey and cricket. 10 boys play only hockey and football. How many boys play all the three games ?

Correct — option (1), 20.

Let x be the number of boys who play all three games.

Step 1 — add the three game totals.
185 + 165 + 160 = 510

Step 2 — see how often each boy is counted in 510.
One game: counted once
Exactly two games: counted twice
All three games: counted three times

Step 3 — the boys who play exactly two games.
Only cricket and football: 40
Only hockey and cricket: 20
Only hockey and football: 10
Total: 70

Step 4 — every boy plays at least one game, so
(one game) + 70 + x = 400

Step 5 — the total of 510 is
(one game) + 2 × 70 + 3x = 510

Step 6 — subtract Step 4 from Step 5.
70 + 2x = 110
2x = 40
x = 20

Step 7 — check the one-game groups.
Only cricket: 185 − 40 − 20 − 20 = 105
Only hockey: 165 − 20 − 10 − 20 = 115
Only football: 160 − 40 − 10 − 20 = 90
105 + 115 + 90 + 70 + 20 = 400 ✓

The idea to remember: "only A and B" counts the boys in exactly those two games; the all-three group sits outside it.

Why the others are wrong

• (2)
  18 — Option (2) does not balance the count. The union is 510 − (70 + 3x) + x = 440 − 2x, and x = 18 gives 404, four more than the 400 boys.

  Each pairwise overlap includes the all-three boys, which is where the 3x comes from. Only x = 20 gives 400.

• (3)
  15 — Option (3) gives 440 − 2 × 15 = 410 boys, ten more than the group.

  The overlaps here are 40 + x, 20 + x and 10 + x, because "only" leaves the all-three boys out of each stated figure. With those overlaps, only x = 20 makes the total 400.

• (4)
  10 — Option (4) gives 440 − 2 × 10 = 420 boys, twenty more than the group.

  With x = 10, only cricket would be 115, only hockey 125 and only football 100. Adding the 70 two-game boys and 10 three-game boys gives 420, not 400.

For three sets, the inclusion–exclusion principle gives the size of the union:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)

Each pairwise overlap n(A ∩ B) includes the members of all three sets. "Only A and B" leaves them out, so n(A ∩ B) = (only A and B) + (all three).

A Venn diagram of three overlapping circles has seven regions inside the circles: three for one set only, three for exactly two sets and one centre for all three.

Key facts

• n(A ∪ B ∪ C) = sum of the three totals − sum of the three pairwise overlaps + the all-three group.
• A pairwise overlap includes the all-three group: n(A ∩ B) = (only A and B) + (all three).
• Here 510 − (70 + 3x) + x = 400, so x = 20.
• Only cricket 105, only hockey 115, only football 90; exactly two games 70; all three 20.
• A three-set Venn diagram has seven regions inside the circles.

Common traps

• Reading "only cricket and football" as everyone who plays both. It excludes the all-three boys; using 40, 20 and 10 as the full overlaps gives 510 − 70 + x = 400 and x = −40, which is impossible.
• Subtracting the two-game boys only once. Each of them appears in two game totals, so each is counted twice in 510.
• Forgetting the all-three group in the one-game check. Only cricket = 185 − 40 − 20 − 20, where the last 20 is the all-three group.

UnlockIAS will link similar questions from RAS Pre 2021 here once that paper is published on this site.

• practice — not a real PYQ

  In a class of 60 students, every student studies at least one of Hindi and English. 35 study Hindi and 30 study English. How many study both ?

  • (a)5
  • (b)10
  • (c)25
  • (d)30
  Answer(1) — n(H ∪ E) = n(H) + n(E) − both: 60 = 35 + 30 − both, so both = 5.

  Option (2) would give 35 + 30 − 10 = 55 students, option (3) is the number who study only English (30 − 5), and option (4) is the number who study only Hindi (35 − 5).

• practice — not a real PYQ

  In a group of 100 people, each speaks at least one of Hindi, English and Rajasthani. 60 speak Hindi, 50 speak English and 40 speak Rajasthani. 10 speak only Hindi and English, 8 speak only English and Rajasthani, and 12 speak only Hindi and Rajasthani. How many speak all three languages ?

  • (a)5
  • (b)10
  • (c)15
  • (d)20
  Answer(2) — The totals add to 150 and exactly-two speakers number 30. One-language + 30 + x = 100 and one-language + 60 + 3x = 150, so 30 + 2x = 50 and x = 10.

  The union 150 − (30 + 3x) + x = 120 − 2x gives 110, 90 and 80 for options (1), (3) and (4), not 100.