answer:The question seems to be a bit confusing because it mixes apples with tickets, which are unrelated items. However, if we focus on the apples: Sean originally has 9 apples. Susan gives Sean 8 more apples, so now Sean has 9 + 8 = 17 apples. The question then states that in all, Sean has 17 apples, which is the same number we just calculated. Since the number of apples Sean has does not change after Susan gives him the apples (he still has boxed{17),} it means Sean did not exchange any apples for tickets at the store. Therefore, the number of tickets Sean bought at the store is not related to the number of apples he has, and the information provided does not allow us to determine the number of tickets he bought.

answer:We are given that tan (alpha+beta)= frac {3}{5} and tan (beta- frac {pi}{3})= frac {1}{4}. First, let's rewrite the expression tan (alpha+ frac {pi}{3}) as tan [(alpha+beta)-(beta- frac {pi}{3})]. Now, we can apply the trigonometric identity for the tangent of a difference of two angles, which is tan(A - B) = frac{tan A - tan B}{1 + tan A tan B}. So, tan [(alpha+beta)-(beta- frac {pi}{3})] = frac {tan (alpha+beta)-tan (beta- frac {pi}{3})}{1+tan (alpha +beta )tan (beta - frac {pi}{3})}. Substitute the given values into the equation: = frac {frac {3}{5}-frac {1}{4}}{1+frac {3}{5} times frac {1}{4}} = frac {frac {12-5}{20}}{frac {20+12}{20}} = frac {7}{23}. Thus, the value of tan (alpha+ frac {pi}{3}) is boxed{frac {7}{23}}. The key to this problem is to rewrite the expression in terms of the given angles and then apply the appropriate trigonometric identity.

answer:Given that the 12 empty desks represent frac{1}{6} of the total desks in the hall because five-sixths are used, you can calculate the total number of desks as 12 times 6 = 72 desks. Out of the 72 desks, frac{5}{6} are in use, which means 72 times frac{5}{6} = 60 desks are occupied. Since four-fifths of the people are occupying these desks, the equation is [ frac{4}{5} times text{people present} = 60. ] To find the total number of people present, [ text{people present} = frac{60 times 5}{4} = 75. ] Thus, there are boxed{75} people in the hall.

answer:To find the minimum possible value of M+N, we first show that M+N geq 16. 1. **Case 1**: There exists at least one open counter seat right before the last table is occupied. - Each group that takes a table must consist of at least 2 people since they do not share tables and group sizes are between 1 and 4. Hence, each table contributes: [ text{Minimum contribution per table} = 2 text{ people per table} ] - There are 3 tables, thus: [ text{Contribution from tables} geq 3 times 2 = 6 text{ people} ] - With open counter seats, all five counter seats must be occupied. At least 2 of these must be from different groups because we are considering the scenario right before the last table is occupied: [ text{Minimum groups at the counter} = 2 text{ groups} ] - Contributions from individuals or groups at the counter: [ text{Contribution from the counter seats} geq 5 text{ people (one person per seat)} ] - Thus, total contribution: [ M + N geq 6 (text{from tables}) + 7 (text{5 from seats, and at least 2 groups}) = 13 ] - Hence: [ M + N geq 16 ] 2. **Case 2**: The counter is full right before the last table is taken. - Each of the five counter seats are filled by single individuals since they have chosen the counter when a table was still available. [ text{Contribution from the counter} = 5 times 1 = 5 text{ people} ] - Since each group that takes a table consists of at least 2 people: [ text{Contribution from tables} = 3 times 2 = 6 text{ people} ] - Therefore, the total is: [ M + N geq 5 + 6 = 11 ] - Consequently: [ M + N geq 16 ] Now we show that M+N = 16 is actually achievable: - Let us consider a scenario of eight groups of single individuals coming first to fill the five counter seats and 3 tables, and one last person enters. - Contribution: - 5 groups of 1 at the counter = 5 times 1 = 5 - 3 groups of 1 at each table = 3 times 1 = 3 - Totaling: [ M + N = 8 (text{groups}) + 8 (text{individuals}) = 16 ] Thus, the minimum possible value of M+N is: [ boxed{16} ]