Question

6 different books are to be distributed among the 10 students in a class. A student can receive multiple books; for example, it's possible that one student receives all 6 of them. How many ways are there to distribute the books: (a) How many ways are there to distribute the books? (b) ... if Alice gets at least one book? (c) ... if only Alice, Bob, and Chloe receive books? (d) ... if only Alice, Bob, and Chloe receive books or only Dan and Emily receive books? (e) ... if no student receives more than one book?

Answer 1

(a) 6 books to be distributed among 10 students, with each student having possibility of receiving multiple books. Hence, each book can be distributed to in 10 different ways. 10 books can be distributed in 10*10*10*10*10*10 ways = 1000000 ways.

(b) If Alice gets atleast 1 book.

Suppose Alice gets no book. Then each of the 6 books have to be distributed among 9 remaining students. Hence, number of ways that Alice gets no book = 9*9*9*9*9*9.

Number of ways Alice gets atleast 1 book = Number of ways books can be distributed among 10 students - Number of ways books can be distributed when Alice gets no book = 10*10*10*10*10*10 - 9*9*9*9*9*9 = 468559

(c) If only Alice, Bob and Chloe receive books then the 6 books have to be distributed between these 3 students. Number of ways in which only Alice, Bob and Chloe receive books = 3*3*3*3*3*3 = 729

(d) Number of ways when only Alice, Bob and Chloe receive books or only Dan and Emily receive books = Number of ways when only Alice, Bob and Chloe receive books + Number of ways when only Dan and Emily receive books = 3*3*3*3*3*3 + 2*2*2*2*2*2 = 793

(e) Can do only 4 subparts of a question