Problem 51 - Algebra and Trigonometry

Question

Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

Step-by-Step Solution

Answer

There are 495 different collections of 4 books you can take from a total set of 12 books.

Step 1: Understand the Problem

You have a collection of 12 books, and you want to select 4 of them to take with you on vacation. This is an instance of a combination problem, because the order in which you select these books does not matter.

Step 2: Applying the Combination Formula

The combination formula is: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where 'n' is total number of books (12), 'k' is the books to select (4), and '!'' represents factorial. So, to calculate the number of combinations of 4 books from a total of 12, plug these numbers into the formula: \[ C(12, 4) = \frac{12!}{4!(12-4)!} \]

Step 3: Solving the Factorials

Factorial is the product of all positive integers up to the number. So: \[ 12! = 12*11*10*9*8*7*6*5*4*3*2*1 \] \[ 4! = 4*3*2*1 \] \[ (12-4)! = 8! = 8*7*6*5*4*3*2*1 \] Plug these factorials into the formula: \[ C(12, 4) = \frac{12*11*10*9*8*7*6*5*4*3*2*1}{4*3*2*1*8*7*6*5*4*3*2*1} \]

Step 4: Simplify the Equation

Simplifying the equation, you can cross cancel the terms from the numerator and denominator: \[ C(12, 4) = \frac{12*11*10*9}{4*3*2*1} \]

Step 5: Calculate the Answer

Finally, multiplying the terms out gives the number of combination: \[ C(12, 4) = 495 \]