Of 12 different books a shelf will hold five. how many different arrangements may be made on the shelf?

Answer

To solve the problem of how many different arrangements can be made on a shelf that holds five books out of twelve different books, we will use the concept of permutations. Here’s a step-by-step solution:

### Step 1: Identify the total number of books and the number of books to arrange
We have a total of 12 different books, and we need to arrange 5 of these books on the shelf.

### Step 2: Use the permutation formula
The number of ways to arrange \( r \) items from \( n \) items is given by the permutation formula:
\[ nPr = \frac{n!}{(n-r)!} \]
In this case, \( n = 12 \) (the total number of books) and \( r = 5 \) (the number of books to arrange).

### Step 3: Substitute the values into the formula
Substituting the values into the permutation formula gives us:
\[ 12P5 = \frac{12!}{(12-5)!} = \frac{12!}{7!} \]

### Step 4: Simplify the factorial expression
To simplify \( \frac{12!}{7!} \), we can expand \( 12! \):
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7! \]
Thus, we can cancel \( 7! \) in the numerator and denominator:
\[ 12P5 = 12 \times 11 \times 10 \times 9 \times 8 \]

### Step 5: Calculate the product
Now we calculate the product:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]
\[ 11880 \times 8 = 95040 \]

### Step 6: Conclusion
The total number of different arrangements that can be made on the shelf is:
\[ \boxed{95040} \]

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