Permutations and Combinations

2.7 Aside: If you are curious about “Permutations and Combinations”

In the previous section we answered the following two problems with ease:

 

Suppose 5 people are to be chosen from 12 and the order in which folk are chosen is not important. How many ways can this be done?

 

Suppose 5 people are to be chosen from 12 for a team and the order in which they are chosen is considered important. In how many ways can this be done?

 

All we had to do was think through what labeling is appropriate in each context.

 

But the traditional mindset on these matters is to give each of these situations different names and to focus on different formulas for solving them.

 

Here is the traditional approach. (And please forget this!)

 

 

COMBINATIONS:

Vaguely, and confusingly, a “combination” is a counting problem in which order chosen does not matter. A typical combination problem would be:

 

Suppose \(r\) people are to be chosen from a pool of \(n\) people and the order in which folk are chosen is not important. How many ways can this be done?

 

We would answer this as:

 

\(r\)  people are to be labeled “chosen.”

\(n-r\) people are to be labeled “not chosen.”

The answer is: \(\dfrac{n!}{r!(n-r)!}\).

 

This is called the “ \(n\) choose \(r\) ” formula and is denoted \(_{n}C_{r}\) on a calculator and in text books.

 

PERMUTATIONS:

Vaguely, and confusingly, a “permutation” is a counting problem in which order chosen does matter. A typical combination problem would be:

 

Suppose  \(r\) people are to be chosen from  a pool of \(n\) people and the order in which folk are chosen is important. How many ways can this be done?

 

We would answer this as:

 

1 person is to be labeled “chosen first.”

1 person is to be labeled “chosen second.”

            …

1 person is to be labeled “chosen \(r\)th.”

\((n-r)\) people are to be labeled “not chosen.”

 The answer is: \(\dfrac{n!}{1!1!\cdots1!(n-r)!}\) .

 

This formula is usually written as \(\dfrac{n!}{(n-r)!}\) and is denoted \(_{n}P_{r}\)  on a calculator and in text books.

 

 

 

Now really do forget these words and these special formulas! Just label!

 

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