Problem 11 - College Algebra

Question

In Exercises \(9-16,\) use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{11} C_{4} $$

Step-by-Step Solution

Answer

The value of \(_{11}C_{4}\) which represents the number of ways you can select 4 items from a set of 11 items is 330.

Step 1: Identifying the values for n and r in the combination

In this problem, \(n = 11\) since there are 11 total items to choose from and \(r = 4\) since 4 items need to be selected.

Step 2: Computing Factorials

Factorial of a number n noted as \(n!\) is calculated as the product of all positive integers less than or equal to n. Thus the factorial of 11 is \(11! = 11*10*9*8*7*6*5*4*3*2*1\), factorial of 4 is \(4! = 4*3*2*1\) and factorial of 7 (since \(11-4 = 7\)) is \(7! = 7*6*5*4*3*2*1\).

Step 3: Calculating Combination

Use the calculated factorials to find the number of ways to choose 4 items from 11. Therefore, \(_{11}C_{4} = \frac{11!}{4!(11-4)!} = \frac{11!}{4!*7!}\) Calculate the above expression to find the value of \(_{11}C_{4}\).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. A factorial, represented by an exclamation mark (!) next to a number, is the product of all positive integers from 1 to that number. For instance, the factorial of 5, written as \( 5! \), is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

This operation is crucial because it helps to determine how many different ways you can arrange or select items. In the given exercise, we see \( 11! \), \( 4! \), and \( 7! \). These are used to calculate combinations, where multiple factorials are applied in dividing scenarios to evaluate multiple choices without regard to order.

Binomial Coefficient

The binomial coefficient is often denoted by \( _{n}C_r \) and is a central topic in combinatorics. This coefficient expresses how many ways you can choose \( r \) items from a set of \( n \) distinct items without considering the order in which they are selected. It answers the fundamental question: "How many combinations are there?"

Mathematically, the binomial coefficient is represented and calculated using the formula:

• \( _{n}C_r = \frac{n!}{r!(n-r)!} \)

Here, \( n! \) is the factorial of the total items, \( r! \) is the factorial of items being chosen, and \( (n-r)! \) is the factorial of the remaining items. This formula allows us to determine combinations efficiently by reducing the large factorial number by the divisions, thus simplifying calculations as seen with \( _{11}C_4 \).

Permutations

Permutations relate closely to combinations but differ due to the importance of order. In permutations, every arrangement of a set is unique depending upon the order of the items. Consider arranging books on a shelf as an illustration; the sequence matters. This contrasts combinations where order is irrelevant, like choosing people for a committee.

To calculate permutations, the formula used is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

While this formula might look similar to the combination formula, it lacks the \( r! \) in the denominator, reflecting the difference that permutations count all possible arrangements as unique occurrences. Combining permutations and combinations is essential for understanding how items can be organized, either with or without consideration of their order.