(19 pts.) A group of 15 engineers from the Engineering Department of a particular IT company decided to organize a summer outing after two years of not being able to travel due to the pandemic. The department consists of 6 females and 9 males. a) (2 pts.) The group decided to appoint an organizing team consisting of lead organizer, assistant lead organizer, and a treasurer. How many possible sets of officers can be formed if only four are interested to become the lead organizer? b) (2 pts.) As recommended by the organizing team, each one of them must be part of a committee. They have identified the following committees: Committee No. of Members Food 5 Sounds 2 Logistics 3 Games 3 Prizes How many ways can the 15 engineers fill up the 5 committees? c) (3 pts.) This is in connection to question (b). If two engineers (one male and one female) have a task to finish and wishes that they can be together in a committee. What is the probability that they will be in one committee? d) (3 pts.) The Food Committee decided to have meeting to plan the food that they will be preparing during their summer outing. The group was able to find a room from their workplace, but it has 8 mounted chairs arrange in one row. How many ways can the members of the Food Committee be arranged in a row of 8 chairs? e) (4 pts.) During their dinner, all of them will sit in a circular table with 15 chairs. How many ways to arrange them such that one group of three female engineers must be seated beside each other and one group of four male engineers must be seated together. The two groups mentioned should not be seated together. f) (2 pts.) One of the male engineers bring out his deck of cards and invited 3 other male engineers to play a Poker game (i.e., a 5-card hand that can assume certain patterns and certain ranks). How many ways full house can be formed given a deck of playing cards? A full house implies that three of the five cards have the same number and the other two cards have the same number. g) (3 pts.) Before they end their summer outing, the lead organizer asks every one of their birthdates. What is the probability at least two of them have the same birthday? (19 pts.) A group of 15 engineers from the Engineering Department of a particular IT company decided to organize a summer outing after two years of not being able to travel due to the pandemic. The department consists of 6 females and 9 males. a) (2 pts.) The group decided to appoint an organizing team consisting of lead organizer, assistant lead organizer, and a treasurer. How many possible sets of officers can be formed if only four are interested to become the lead organizer? b) (2 pts.) As recommended by the organizing team, each one of them must be part of a committee. They have identified the following committees: Committee No. of Members Food 5 Sounds 2 Logistics 3 Games 3 Prizes How many ways can the 15 engineers fill up the 5 committees? c) (3 pts.) This is in connection to question (b). If two engineers (one male and one female) have a task to finish and wishes that they can be together in a committee. What is the probability that they will be in one committee? d) (3 pts.) The Food Committee decided to have meeting to plan the food that they will be preparing during their summer outing. The group was able to find a room from their workplace, but it has 8 mounted chairs arrange in one row. How many ways can the members of the Food Committee be arranged in a row of 8 chairs? e) (4 pts.) During their dinner, all of them will sit in a circular table with 15 chairs. How many ways to arrange them such that one group of three female engineers must be seated beside each other and one group of four male engineers must be seated together. The two groups mentioned should not be seated together. f) (2 pts.) One of the male engineers bring out his deck of cards and invited 3 other male engineers to play a Poker game (i.e., a 5-card hand that can assume certain patterns and certain ranks). How many ways full house can be formed given a deck of playing cards? A full house implies that three of the five cards have the same number and the other two cards have the same number. g) (3 pts.) Before they end their summer outing, the lead organizer asks every one of their birthdates. What is the probability at least two of them have the same birthday?