Ordered selections

The multiplication principle with boxes, ordered selections with repetition (n to the power r) and without repetition (nPr), permutations of a whole set, and dealing with restrictions first.

The multiplication principle

Draw one box for each position and write the number of choices in each box. Then multiply. A three-letter code from A to Z, for example, is $\boxed{26}\,\boxed{26}\,\boxed{26} = 26^3$.

With and without repetition

Repetition allowed

Each of the $r$ positions has all $n$ choices:

$$n^r$$

No repetition

Each choice uses up one object:

$${}^nP_r = n(n - 1)\cdots(n - r + 1) = \frac{n!}{(n - r)!}$$

${}^nP_r$ is the number of permutations of $r$ objects chosen from $n$. Arranging the whole set ($r = n$) gives ${}^nP_n = n!$.

Example 1●

How many four-digit PINs are there? How many have no repeated digit?

With repetition: $10^4 = 10\,000$.

Without: $\boxed{10}\,\boxed{9}\,\boxed{8}\,\boxed{7} = {}^{10}P_4 = 5040$.

Difficulties first

Example 2●●

How many three-digit numbers can be formed from $0, 1, 2, 3, 4, 5$ with no repeated digit? How many of them are even?

$\boxed{5}\,\boxed{5}\,\boxed{4} = 100$

The first digit cannot be 0.

Ending in $0$: $\boxed{5}\,\boxed{4}\,\boxed{1} = 20$. Ending in $2$ or $4$: $\boxed{4}\,\boxed{4}\,\boxed{2} = 32$.

Even: the last digit is 0, 2 or 4. Split on whether it is 0, because 0 also affects the first digit.

Even numbers: $20 + 32 = 52$.

Flashcards

Ordered, with repetition

$n^r$

${}^nP_r$

$\dfrac{n!}{(n - r)!}$

Arranging $n$ different objects

$n!$

Restrictions

Fill the restricted positions first.

Practice

How many four-digit PINs are possible?
How many four-digit PINs have no repeated digit?
Evaluate ${}^6P_3$.
${}^nP_r = $
In how many ways can $6$ people stand in a line?
How many three-digit numbers can be formed from $0, 1, 2, 3, 4, 5$ with no repeated digit?
How many of those are even?
In how many ways can a gold and a silver medal go to $8$ runners?
A code of $3$ letters followed by $2$ digits, repetition allowed. The number of codes is:
With restrictions, the boxes you should fill first are: