Back to Explorer Missions
Ethan has 12 books, and he wants to donate all of them to three local libraries: Central, Westside, and Eastside. He doesn’t necessarily want to distribute the books equally, but he wants each library to receive at least one book.
In how many ways can Ethan distribute the books to Central, Westside, and Eastside libraries?
Let’s approach this systematically:
1. We need to consider all possible distributions where Central Library receives 1 to 10 books (if it receives 11, we can’t give at least 1 to each of the other libraries).
2. For each case, we’ll count how many ways we can distribute the remaining books between Westside and Eastside:
– Central receives 1 book: 11 books left, 10 ways to distribute (1,10), (2,9), …, (10,1)
– Central receives 2 books: 10 books left, 9 ways to distribute
– Central receives 3 books: 9 books left, 8 ways to distribute
– …
– Central receives 10 books: 2 books left, 1 way to distribute (1,1)
3. The total number of ways is the sum of all these possibilities:
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55
4. We can verify this using the formula for the sum of the first n positive integers:
n(n+1)/2, where n = 10
10(11)/2 = 55
Therefore, Ethan can distribute the 12 books to the three libraries in 55 different ways.
Remember, at QMAK, we don’t just teach; we empower. We don’t just inform; we inspire. We don’t just question; we act. Become a Gold Member, and let’s unlock your child’s full potential, one question at a time.