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Emma has a small bookshelf with space for 6 books in a row. She has 4 fiction books (each marked with an F) and 2 non-fiction books (each marked with an N). Emma wants to arrange these books on the shelf.
How many different ways can Emma arrange the 6 books on the shelf?
Note: Switching two fiction books doesn’t count as a new arrangement, since all the fiction books look the same. The same applies to the non-fiction books.
Let’s solve this problem step-by-step using the concept of combinations:
1. We can think of this problem as choosing positions for the non-fiction books (N). Once we place the N books, the F books will fill the remaining spots.
2. We need to choose 2 positions out of 6 for the N books. This can be calculated using the combination formula:
C(6,2) = 6! / (2! * 4!) = (6 * 5) / 2 = 15
3. Here’s how we can break it down:
– N N F F F F : 1 way
– N F N F F F : 4 ways
– N F F N F F : 3 ways
– N F F F N F : 2 ways
– N F F F F N : 1 way
Total: 1 + 4 + 3 + 2 + 1 = 11 ways
4. We can verify this by listing all possible arrangements:
NNFFFF, NFNFFF, NFFNFF, NFFFNF, NFFFFN,
FNFNFF, FNFFNF, FNFFFN,
FFNFNF, FFNFFN,
FFFNNF
Therefore, there are 15 different ways Emma can arrange the 6 books on the shelf.
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