Understanding the Problem: Arranging Books on a Shelf

When it comes to arranging items such as books on a shelf, particularly when there are duplicates among them, the concept of permutations of a multiset becomes essential. This article aims to guide you through the process of calculating the number of possible arrangements using the given example: 4 Math books, 3 Science books, 2 Filipino books, and 1 English book, totaling 10 books. We will delve into the mathematical formula and provide detailed calculations for clarity and understanding.

The Formula for Permutations of a Multiset

To determine the number of ways to arrange a set of items where some items are identical, we use the formula for permutations of a multiset:

( frac{n!}{n_1! times n_2! times cdots times n_k!} )

where ( n ) is the total number of items, and ( n_1, n_2, ldots, n_k ) are the counts of each type of identical item.

Applying the Formula to the Given Problem

In our problem, we have:

4 Math books (M) 3 Science books (S) 2 Filipino books (F) 1 English book (E)

Total number of books, ( n ), is 10. We use the formula:

( text{Number of arrangements} frac{10!}{4! times 3! times 2! times 1!} )

Breaking Down the Calculation

First, we calculate the factorials:

10! 3,628,800 4! 24 3! 6 2! 2 1! 1

Substituting these values into our formula:

( text{Number of arrangements} frac{3,628,800}{24 times 6 times 2 times 1} )

Next, we perform the multiplication in the denominator:

24 × 6 144 144 × 2 288 288 × 1 288

Finally, we divide the numerator by the denominator:

( frac{3,628,800}{288} 12,600 )

Therefore, the total number of ways to arrange the 10 books is 12,600.

Alternative Methods for Arrangement

Let's explore an alternative method to verify our result. Using the formula for arranging items with repetitions:

( frac{10!}{4! times 3! times 2! times 1!} )

Perform the factorial calculations:

10! 3,628,800 4! 24 3! 6 2! 2 1! 1

Divide the results:

( 3,628,800 ÷ 24 × 6 × 2 × 1 12,600 )

This confirms our previous calculation.

Another Perspective: Arranging Books by Sampling

Another approach is to consider each step of placing the books. We have 10 books in total:

1. Place the English book: 10 options

2. Place the Filipino books: ( frac{9 times 8}{2!} 36 ) options (correcting for the order of the 2 books)

3. Place the Science books: ( frac{7 times 6 times 5}{3!} 70 ) options (correcting for the order of the 3 books)

4. Place the Math books: The remaining spots are automatically occupied by the 4 Math books

Combining these steps, we get:

( 10 times frac{9 times 8}{2!} times frac{7 times 6 times 5}{3!} 10 times 36 times 70 12,600 )

Alternatively, using combinations:

( binom{10}{1} binom{9}{2} binom{7}{3} binom{4}{4} 10 times 36 times 35 times 1 12,600 )

Conclusion

Understanding permutations of a multiset is key to solving problems involving the arrangement of items with duplicates. Applying the formula and breaking down the problem into smaller, manageable steps helps in calculating the total number of arrangements accurately. Whether you use factorials, combinations, or step-by-step sampling, the result remains the same: 12,600 possible arrangements for our collection of 10 books.