Lara is turning 9 and is having a birthday party! (a) There are 27 kids in Lara's class (including herself). Her best friends Fred, Asha and Adam are in her class. For her birthday, she would like to invite 7 children from her class (not counting herself). She invites Fred, Asha and Adam and 4 other children. How many differ- ent combinations are possible? (b) Each of the kids at the party (including the host Lara) is given one of the eight colored T- shirts that Lara's father has purchased for the kids (green, blue, orange, red, yellow, pink, 2x purple). In how many ways can the T-shirts be distributed to the children? (c) There is a big bucket with purple, blue, green, yellow, orange and red lollies (hundreds of each color). The green lollies are the tastiest. One after the other, the children throw a colored dice (with six faces corresponding to the colors above), until each child has rolled the dice once. What is the probability that more than 3 children get a green lolly? (d) The kids (including the host Lara) line up in alphabetical order. Lara's mother comes with a bucket that has cardboard cut-out number in it; it contains exactly one copy of each of the 10 numbers 0 to 9. The children line up, and each child picks a cardboard cut-out randomly and holds it up above his or her head. What is the probability that the number that is formed is 1,234,567? What is the probability that the number that is formed is greater than or equal to 4,000,000? (e) Fred, Asha and Adam have brown hair; all other kids have blond hair. For a particular game, Lara's father selects a team of 3 kids with brown hair and 2 kids with blond hair. In how many different ways can he put together this team? (f) The seven guests are called Felix, Lana, Adam, Asha, Fred, John and Eric. All children (including the host Lara and including the guest Lana) play a fun game where they make new words out of their first name initials. How many distinct six-letter words that have two 'A' and two 'F' but only one 'L’ can they make out of their initials? How many distinct six-letter words that have two 'A' and two 'F' (and no restrictions on 'L') can they make out of their initials? (The initial of each person can only be used at most once, that is, 'F', 'L' and 'A' at most twice and 'J' and 'E' at most once; a word is here defined as a sequence of letters, regardless of whether it makes sense in the English language.)