Can you explain this on paper for a better understanding?

Chapter 1: Permutations In how many ways the word Mathematics can be arranged ? Consider the word BASKETBALL: a. How many permutations are there? b. How many permutations begin with the letter K? c. How many permutations have the two L's together? How many arrangements of the word ALGORITHM with vowels are together . 4. The number of ways that the 12 members of a baseball team can be lined up if Zoe, Izaak, and Ally must all be together is Note: It is assumed that no 2 members of the team share the same name. a) 21 772 800 b) 79 833 600 c) 2 177 280 d) 10 886 400 5.How many ways can 5 people be arranged in a line for a photograph if there are 7 people in a group to pick from? 2007! 6..The value of the expression 2005! is a) not possible b) 4 026 042 c) approximately 1. d) none of the above. 6. Find the number of arrangements for - a) Calculus b) MANIACAL c) ALIEN when vowels are always together