Permutations for Counting Techniques

$$ \large \require{AMSsymbols} \displaystyle \begin{align} ^n P_r &= \frac{n!}{(n-r)!} \\ &= \overbrace{n \times (n-1) \times (n-2) \times \cdots }^{r} \end{align} $$

Example

$ ^{10} P _3 = \overbrace{10 \times 9 \times 8}^{3} = 720 $

$ \displaystyle ^{10} P _3 = \frac{10!}{(10-3)!} = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 = 720 $

Question 1

Evaluate the following.

(a) $ ^9 P _2 $

$ =9 \times 8 = 72 $

(b) $ ^5 P _1 $

$ =5 $

$ \displaystyle =\frac{6!}{(6-0)!} = 1 $

(d) $ ^4 P _4 $

$ =4 \times 3 \times 2 \times 1 = 4! = 24 $

(e) $ ^5 P _6 $

$ \text{undefined} $

(f) $ ^5 P _{-1} $

$ \text{undefined} $

Question 2

Find the value of $ n $, if $ ^6 P_n = 120 $.

$ \begin{align} 6 \times 5 &= 30 \\ 6 \times 5 \times 4 &= 120 \\ ^6 P_3 &= 120 \\ \therefore n &=3 \end{align} $

Question 3

The digits $ 1, \ 2, \ 3$ and $ 4 $ are each written on a card and then placed in a box.

(a) A card is chosen, read and replaced. A second card is chosen and read. How can many possible two digit-numbers be chosen?

$ 4 \times 4 = 15 $

(b) A card is chosen, read and not-replaced. A second card is chosen and read. How can many possible two digit-numbers be chosen?

$ ^4 P_2 = 4 \times 3 = 12 $

Question 4

How many ways are there of choosing three books from a shelf of ten and reading them in order?

$ ^{10} P_3 = 720 $ ways

Question 5

In how many ways can the letters of the word TODAY be arranged if they are used once only and taken:

(a) three at a time?

$ ^5 P_3 = 60 $

(b) four at a time?

$ ^5 P_4 = 120 $

$ ^5 P_5 = 120 $

Issue: *
Your Name: *
Your Email: *

Details: *