Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

Verified step by step guidance

1. Identify the problem as a combination problem where order does not matter, and you want to find the number of ways to choose 4 books out of 12.
2. Recall the formula for combinations (nCr), which is given by: $\mathrm{nCr}=\frac{n!}{r!\; (n-r)!}$, where $n$ is the total number of items and $r$ is the number of items to choose.
3. Substitute the given values into the formula: $n=\; 12$ and $r=\; 4$, so the expression becomes $\backslash binom\{12\}\{4\}\; =\; \backslash frac\{12!\}\{4!\; (12-4)!\}$.
4. Calculate the factorial expressions in the numerator and denominator separately, but do not compute the final numerical value yet. Remember that factorial means the product of all positive integers up to that number.
5. Simplify the expression by canceling common factorial terms in the numerator and denominator to make the calculation easier before finding the final number of combinations.

Key Concepts

Combination Formula (nCr)

The combination formula, denoted as nCr, calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is essential for counting selections where order does not matter.

Factorials

A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations and combinations to calculate the total number of arrangements or selections.

Distinguishing Between Combinations and Permutations

Combinations count selections where order does not matter, while permutations count arrangements where order matters. In this problem, since the order of books does not affect the collection, combinations are used to find the number of different groups of books.