How many ways can 9 books be arranged on a shelf so that 5 of the books are always together?

As Robert Shore mentions in his comment, books of a particular type (calculus/DM) are probably being considered as indistinguishable.

The relevant calculation here is instead $${12\choose3}-{10\choose1}=220-10=210$$

where we choose $3$ of $12$ positions to belong to DM books for the total case and subtract $10\choose1$ for the case where they all are together (choose $1$ out of $10$ positions for the 3 DM books to be placed together).

Part (b) can be solved by considering that we can place $x$ calculus books between the first and middle DM book and $(9-x)$ calculus books between the middle and last DM book. $x$ can take the values $0,1,2...9$ ($10$ different values) and thus the answer here is $10$ different arrangements.

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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

• a,b,c,d,e,f,g,h,i
• a,c,b,d,e,f,g,h,i
• b,a,c,d,e,f,g,h,i
• b,c,a,d,e,f,g,h,i
• c,a,b,d,e,f,g,h,i
• c,b,a,d,e,f,g,h,i

g,h,i,j,d,e,f

• g,h,i,a,b,c,d,e,f
• g,h,i,a,c,b,d,e,f
• g,h,i,b,a,c,d,e,f
• g,h,i,b,c,a,d,e,f
• g,h,i,c,a,b,d,e,f
• g,h,i,c,b,a,d,e,f

Answer: 30,240

362,880 - 30,240 = 332,640

Answer: 332,640