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38 votes
38 votes
A kindergarten teacher has five books to distribute to 20 children in her class.

(a)
How many ways are there for her to distribute the books if they are all the same and no child gets more than one?
(b)
How many ways are there for her to distribute the books if they are different and no child gets more than one? If Charlie gets Green Eggs and Ham and Amanda gets The Cat in the Hat, that is a different distribution from one in which Amanda gets Green Eggs and Ham and Charlie gets The Cat in the Hat.
(c)
How many ways are there for her to distribute the books if they are all the same and there is no restriction on the number of books that can be given to any child?

User Zbyszek
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2 Answers

24 votes
24 votes

Final answer:

a. The number of ways to distribute 5 identical books among 20 children is 42,504. b. The number of ways to distribute 5 distinct books among 20 children is 3,840. c. The number of ways to distribute 5 identical books among 20 children with no restrictions on the number of books each child can receive is 3,656,158,440,062,976.

Step-by-step explanation:

a. In this case, we need to distribute 5 books among 20 children. Since each book is the same and no child can receive more than one book, this is equivalent to distributing 5 identical objects into 20 distinct boxes. The number of ways to do this is given by the stars and bars formula, which is C(n+r-1, r-1) where n is the number of objects to be distributed (5 books) and r is the number of boxes (20 children). So, the number of ways is C(5+20-1, 20-1) = C(24, 19) = 42,504.

b. In this case, since the books are different and no child can receive more than one book, we need to find the number of ways to distribute 5 distinct books into 20 distinct boxes. This can be calculated using the permutation formula, which is P(n, r) = n! / (n-r)!, where n is the total number of objects and r is the number of objects to be selected. So, the number of ways is P(5, 20) = 20! / (20-5)! = 20! / 15! = 3,840.

c. In this case, there are no restrictions on the number of books that can be given to any child. So, each child can receive any number of books from 0 to 5. This means there are 6 options (0, 1, 2, 3, 4, or 5) for each child, and since there are 20 children, the total number of ways to distribute the books is 6^20 = 3,656,158,440,062,976.

User Ellery Newcomer
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3.3k points
10 votes
10 votes

Answer:

(a) There are 20 ways for her to distribute the books if they are all the same and no child gets more than one. Each of the 20 children can receive one of the five books.

(b) There are 100 ways for her to distribute the books if they are different and no child gets more than one. Each of the 20 children can receive one of the five different books, so there are 5x4x3x2x1 = 100 possible ways for her to distribute the books.

(c) There are 3,628,800 ways for her to distribute the books if they are all the same and there is no restriction on the number of books that can be given to any child. This is because each child can receive any number of books from 0 to 5. Therefore, there are 6x6x6x6x6 = 6^5 = 3,628,800 possible ways for her to distribute the books.

Step-by-step explanation:

User Mario Aguilera
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2.9k points