Can someone please help me? I've been trying to do this for hours:
You are part of a reading group. Each member will read a book from 1 out of 4 different authors(W,X,Y,Z). In how many ways is it possible to select 25 books from theses authors so there are at least 5 books from each author and, at maximum, 6 books from author X?
obs: the titles of the 25 books don't matter, only who the author is
Hint
Suppose that you have the following two problems:
What is the number of positive integer solutions to $x_1 + x_2 + x_3 = n \in \mathbb{Z^+}$, where each of the 3 variables must be $\geq k \in \mathbb{Z^+}$ and $n > 3k$
What is the number of non-negative integer solutions to $x_1 + x_2 + x_3 = (n-3k).$
What is the relationship between the answers to these two problems?
As the titles of the books don't matter but only authors, this translates to finding number of ways to pick different number of books from authors with given restrictions. As you need at least $5$ books of each author ($20$ books across $4$ authors), you are only left with $5$ books to pick.
Take two cases,
i) $5$ books from author $X$ - So you need to pick $5$ books from $3$ remaining authors
ii) $6$ books from author $X$ - you need to pick $4$ books from $3$ remaining authors.
You can use stars and bars method to find number of ways for both cases $1$ and $2$.
