Permutations for Counting Techniques

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

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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)     \( ^5 P _1 \)

\( =5 \)

(c)     \( ^6 P _0 \)

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

(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 \( ^6 P_n = 120 \).

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

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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?

\( ^5 P_4 = 120 \)

(c)     five at a time?

\( ^5 P_5 = 120 \)

 

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