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We can categorize people into four groups: 1. Men who smoke (M_S) 2. Men who drink (M_D) 3. Women who smoke (W_S) 4. Women who drink (W_D) We will create a two-way table to represent these groups and their intersections (i.e. smoking and drinking): | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | | | | Total Women | | | |

Fill in the table with given data: - 18 men smoke but do not drink (M_S∩M_D') - 13 men drink but do not smoke (M_S'∩M_D) - 9 women are non-smokers (W_S') - 7 women drink but do not smoke (W_S'∩W_D) | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | 7 | | | Total Women | | 13 | 33 | |-----------|-----------|-----------|-----------| | Total Men | 18 | | 37 | The table is partially filled now.

Now, we will use the Principle of Inclusion-Exclusion (PIE) to fill the remaining values in the table. Calculate Total Men who smoke (M_S): Men who smoke but do not drink = 18 Therefore, M_S = 18. Calculate Total Men who drink (M_D): M_D = Total Men who drink - Men who drink but do not smoke M_D = 37 - 13 M_D = 24 Calculate Total Women who smoke (W_S): W_S = Total Women - Women who are non-smokers W_S = 33 - 9 W_S = 24 Calculate Total Women who drink (W_D): W_D = Total Women who drink - Women who drink but do not smoke W_D = 33 - 7 W_D = 26 Update the table: | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | 7 | | | Total Women | 24 | 26 | 33 | |-----------|-----------|-----------|-----------| | Total Men | 18 | 24 | 37 | Now there is only one cell left which is for people who both drink and smoke.

According to the problem, there are a total of 36 teetotalers in the group. Based on what we have found so far, sum of non smoking men and women = 18 + 9 = 27. Let x be the number of people who both drink and smoke: Total teetotalers = Total men + Total women - People who drink and smoke - People who neither drink nor smoke 36 = 37 + 33 - x - 27 x = 24 Now, our table is complete: | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | 7 | | | Total Women | 24 | 26 | 33 | |-----------|-----------|-----------|-----------| | Total Men | 18 | 24 | 37 | |-----------|-----------|-----------|-----------| | Both | | | 24 | So, at most, there are 24 people who both drink and smoke in the group.