Problem 5 - Thinking Mathematically

Question

In Exercises 5-20, use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{6} C_{5}\)

Step-by-Step Solution

Answer

\({ }_{6} C_{5} = 6\)

Step 1: Identify the values of n and r

Here, n is the total number of items, which is 6, and r is the number of items to choose, which is 5.

Step 2: Substitute the values into the formula

The combination formula is \({ }_{n} C_{r} = \frac{n!}{r!(n-r)!}\). Substituting the values gives \({ }_{6} C_{5} = \frac{6!}{5!(6-5)!}\).

Step 3: Compute the factorial of the numbers

Compute 6!, 5! and (6-5)! respectively. 6! = 6*5*4*3*2*1 = 720, 5! = 5*4*3*2*1 = 120, and (6-5)! which simplifies to 1! = 1

Step 4: Substitute the factorials into the formula

Substituting the factorials into the formula gives \({ }_{6} C_{5} = \frac{720}{120 * 1}\)

Step 5: Simplify the equation

Dividing 720 by 120 by 1 gives 6. Therefore, \({ }_{6} C_{5} = 6\)

Key Concepts

Combination Formula

The combination formula is a powerful tool in combinatorics and is used to determine the number of ways to choose a subset of items from a larger set, without considering the order of selection. This is useful when you want to know how many ways you can select a specific number of items from a larger group. The general formula for combinations is denoted as \(_{n}C_{r} = \frac{n!}{r!(n-r)!}\), where:

• \(n\) is the total number of items in the set.
• \(r\) is the number of items to choose.
• \(n!\), \(r!\), and \((n-r)!\) are the factorials of \(n\), \(r\), and \(n-r\) respectively.

This formula helps in figuring out all the possible ways *without repetition* where the sequence does not matter.

Factorial

Factorials are foundational in understanding combinations and permutations. The factorial of a number \(n\), denoted as \(n!\), represents the product of all positive integers from 1 to \(n\).For instance,

• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• \(1! = 1\)

Factorials are key because they help calculate the number of permutations (or arrangements) and combinations (selection of items). Each step of the factorial reduces the problem by one possibility!

Combinations and Permutations

Combinations and permutations are two separate concepts often compared in combinatorics. While they both relate to ways of organizing items, they differ significantly in whether order matters.

• Combinations disregard order. For example, the group \(\{a, b, c\}\) is identical to \(\{c, b, a\}\) when considering combinations. For choosing \(r\) items from \(n\) items, use the combination formula \[n_{C}r\].
• Permutations take order into account. Here, \(\{a, b\}\) is different from \(\{b, a\}\). Permutations involve more arrangements than combinations and are calculated using \[n_{P}r = \frac{n!}{(n-r)!}\].

Understanding when to use combinations versus permutations is crucial in solving probability and arrangement problems effectively.

Mathematical Notation

Mathematical notation streamlines complex calculations and provides a universal language among mathematicians. The notation \(nCr\) and \(nPr\) are specifically used for combinations and permutations respectively, reflecting the number of ways to choose \(r\) items from \(n\) items.

• The exclamation mark – \(!\) signifies a factorial, which is a crucial part of these calculations.
• Parentheses in formulas help in defining the order of operations clearly.

This structured approach in written format facilitates less room for error and enhances comprehension when learning or teaching mathematical concepts. Recognizing and understanding these symbols is vital for progressing in mathematics.