There are only 5 places on the shelf. You have 7 books to choose from. We will ignore the order of the books on the shelf. The first place can be filled from a choice of 7 books, the next place from 6, the next place from 5, the next from 4, and the last of the 5 places from 3 books. So the number of ways of choosing the 5 is found from 7 * 6 * 5 * 4 * 3 = 2520 Part (a) We have 9 books which I'll call book a, book b, book c, ..., book h, book i Let's say we want books a through c to always stick together in any particular order. We can pull books a,b,c out of the group on a temporary basis for now. Replace them with book j. The position of book j will represent books a,b,c in any order. The 9 books drop to 9-3 = 6 after taking books a,b,c out, but then the count bumps up to 6+1 = 7 after adding book j. Arrange the 7 books and you should find there are 7! = 7*6*5*4*3*2*1 = 5040 permutations. The order matters. You can alternatively use the nPr formula with n = 7 and r = 7. That 5040 describes sequences involving book j. Wherever you see book j, replace it with some permutation of a,b,c. For example, if we had the sequence j,d,e,f,g,h,i then it could represent any of the following 6 items
g,h,i,j,d,e,f could represent any of the following
Answer: 30,240 ----------------------------------------------------------- Part (b) There are 9! = 9*8*7*6*5*4*3*2*1 = 362,880 different ways to arrange all of the books regardless if 3 particular books stick together or not. We found earlier there are 30,240 ways to arrange the books so that 3 stick together. Subtract those two values to get362,880 - 30,240 = 332,640 Answer: 332,640 |