In a group of 100 children, 64 children like to play cricket, 53 children like to play football and 20 children like to play both cricket and football. How many children do NOT like to play cricket or football?

Correct — A, 3. Use the addition rule for two overlapping groups. The number who like at least one of the two games is the number who like cricket plus the number who like football minus the number counted twice in both, that is 64 + 53 − 20 = 97. Everyone else likes neither, so 100 − 97 = 3 children. The subtraction of 20 is the point of the question: adding 64 and 53 gives 117, which already exceeds the whole group of 100, and the excess exists only because the twenty who play both have been counted once in each total. Removing that double count brings the figure down to 97 and leaves three children outside both circles.

• (b)5 — Five would require the overlap to be 22 rather than 20, or one of the two group totals to be smaller by two. The printed figures give 97 in the union and no other value.
• (c)7 — Seven corresponds to an overlap of 24. It is a plausible-looking small number with nothing in the stem to support it.
• (d)9 — Nine would need an overlap of 26. This is the option for a candidate who guesses that a few more children must be outside without doing the subtraction.

For two sets the rule is n(A or B) = n(A) + n(B) − n(A and B), and the count outside both is the total minus that union. It is worth being able to break the group down completely: cricket only is 64 − 20 = 44, football only is 53 − 20 = 33, both is 20, and neither is 3 — and those four numbers add back to 100, which is the check to run before writing the answer down.

The signal that a double count is present is that the two group totals sum to more than the whole. Here 64 plus 53 is 117 against a group of 100, so at least seventeen children must be in both; the stem says twenty, which leaves room for three outside. That inequality reasoning is worth keeping, because it also tells you when a set of figures is impossible — an overlap smaller than seventeen would have been inconsistent with a group of only a hundred.

• n(A or B) = n(A) + n(B) − n(A and B), so 64 + 53 − 20 = 97 like at least one game.
• Those liking neither number 100 − 97 = 3.
• The four disjoint groups are cricket only 44, football only 33, both 20, and neither 3, adding to 100.
• When two group totals exceed the whole, the excess is a lower bound on the size of the overlap.
• Here 64 + 53 − 100 = 17, so at least seventeen children had to like both.

• Subtracting the overlap twice and reaching 77 in the union.
• Answering 97, the union, when the question asks for those outside it.
• Forgetting to check that the four disjoint parts add back to the stated total.

Asked as the plainest possible two-set inclusion-exclusion item; the only decision is whether to add or subtract the overlap, and the four options are all small numbers.

• practice — not a real PYQ

  In a group of 80 people, 45 read the newspaper, 38 watch television news and 12 do neither. How many do both?

  • (a)9
  • (b)12
  • (c)15
  • (d)20

  Answer(c) 15 — the union is 80 − 12 = 68, so the overlap is 45 + 38 − 68 = 15.

• practice — not a real PYQ

  In a class of 60, 35 play chess and 30 play carrom, and every student plays at least one of the two. How many play both?

  • (a)5
  • (b)10
  • (c)15
  • (d)25

  Answer(a) 5 — with no one outside, the union is 60, so the overlap is 35 + 30 − 60 = 5.