Problem 67 - Algebra and Trigonometry

Question

Explain the difference between a permutation and a combination. Give an example to illustrate your explanation.

Step-by-Step Solution

Answer

Permutations consider the order of arrangement; combinations do not. For 5 objects taken 3 at a time, there are 60 permutations and 10 combinations.

Step 1: Define Permutation

A permutation is an arrangement of all or part of a set of objects in a specific order. The order of arrangement is crucial in permutations. The number of permutations of n objects taken r at a time is given by the formula: \[P(n, r) = \frac{n!}{(n-r)!}\]

Step 2: Define Combination

A combination is a selection of all or part of a set of objects without regard to the order of the objects. The order does not matter in combinations. The number of combinations of n objects taken r at a time is given by the formula: \[C(n, r) = \frac{n!}{r!(n-r)!}\]

Step 3: Example of Permutation

Consider selecting 3 students out of 5 (A, B, C, D, E) to form a line. The number of ways to arrange these 3 students is a permutation since the order matters. The total permutations are: \[P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!}= 60\]

Step 4: Example of Combination

Consider selecting 3 students out of 5 (A, B, C, D, E) to form a committee. The number of ways to choose these 3 students is a combination since the order does not matter. The total combinations are: \[C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = 10\]

Step 5: Summarize Difference

The key difference between permutations and combinations is whether the order of selection matters. If the order matters, it is a permutation. If the order does not matter, it is a combination.

Key Concepts

Permutations

Permutations are arrangements of objects where the order is important. Think of permutations like organizing books on a shelf. If you move one book, it changes the entire arrangement. Mathematically, permutations of n objects taken r at a time are represented by the formula: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: ax: axcontra axcy axicy axmn axmn axmn axmn axmn axmn axaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxax ax: ax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax: axax: ax: ax: ax axaxaxaxaxaxax

Combinations

Combinations involve selecting items where order doesn't matter. For example, if you are picking 3 fruits from a basket of bananas, apples, and oranges, you care about which fruits you pick, not the order in which you pick them. Mathematically, the formula for combinations of n objects taken r at a time is: Here's a breakdown:

• The numerator is the factorial of the total items, n.
• The denominator has the factorial of chosen items, r, and the factorial of remaining items (n-r).

For 3 selections out of 5 students, the calculation would be:

axay: ax: ax: ax:axax: ax: ax: ax:axax: ax: ax: ax: ax

Factorial

A factorial, denoted by the symbol !, involves multiplying a series of descending natural numbers. For instance, 5! is calculated as: factorial (denoted by !) is a very important part of permutations and combinations. It can be calculated as: axayax:axayaxayaxayaxayaxayaxayaxayaxayaxayaxay: axayaxI having account axayaxnumbersay UNTIL bely by the and formula AND every number until 1. Conta<|vq_3927|> axayaxayaxayaxayaxayaxayaxayaxayklarckayTempo axkayarnaxaxayaxayaxayaxayaxaxayaxaydThe factorial of a given quantity n, written as n, e.g 5! is: 5*4*3*2*1 = 120. Factorials check important for permutations scaffolds combinaxayaxayaxayaxay

Order of Arrangement

Order of arrangement is crucial in understanding the difference between permutations and combinations. If the sequence in which the items are arranged affects the outcome, as in permutations, then the arrangement matters. For example,

• In the word 'CAT,' changing the order to 'TAC' changes the meaning.
• When arranging books on a shelf, the order impacts the final look.

However, in combinations, the order is irrelevant. Picking 3 fruits—apple, banana, and orange—is the same as picking orange, banana, and apple.

Understanding when order matters helps decide the method—permutations or combinations—appropriate for solving a problem.