CAT 2022 Slot 1 · Quant · Question 17

Call the number who like all three x.

For the maximum, note that anyone liking all three must in particular like lemonade.

So x can never exceed 52, the size of the smallest group.

That bound is reachable: let all 52 lemonade-likers also like coffee and tea, which is possible since 73 and 80 both exceed 52.

So the maximum is 52.

For the minimum, add the three group sizes: 73+80+52=205.

Each of the 100 students can be counted at most three times in that total, so the excess over the class size must be absorbed by overlaps.

The sharpest bound is x\ge 73+80+52-2\times 100, since every student counted once is covered by the first 100 and every student counted twice by the second.

That gives x\ge 205-200=5.

This too is reachable, with 5 students liking all three and the rest arranged to like exactly two or fewer.

So the difference is 52-5=47 — the third option.

Shortcut. Two standard bounds settle it: the triple overlap is at most the smallest set, and at least ∑-2n for three sets over n people. Here that is 52 and 205-200=5, so the answer is 52-5 with no Venn bookkeeping.