Permutations: Quick Revision

A brief note on Permutations. Read it before a test. Try each question at the end before you open its answer.

The formulas

Also $0! = 1$, ${}^nP_0 = 1$ and ${}^nP_r = n \times {}^{n-1}P_{r-1}$.

Conditions

• Together. Glue the objects into one block. Multiply by the orders inside it, unless they are alike.
• Apart. All arrangements minus the together ones.
• Fixed places. Fill the restricted places first.
• Digits. A number cannot start with $0$. Split into cases when $0$ meets another condition.
• Rank. Count the words before it, one place at a time, then add $1$.

A worked example

How many arrangements of BALLOT have the two L's side by side?

Step 1. All arrangements. $6$ letters with L twice: $\dfrac{6!}{2!} = 360$.

Step 2. Glue the L's. The block and B, A, O, T make $5$ units, all different: $5! = 120$.

Step 3. Inside the block. The L's look alike, so there is only one order.

Answer: $120$ of the $360$ arrangements keep the L's together.

Practice with new numbers

A program to try

The program lists the arrangements of BALLOT and keeps those where the L's stand together.

Mistakes to avoid

• $n^r$ when objects cannot repeat. Use ${}^nP_r$.
• $\dfrac{n!}{r!}$ for ${}^nP_r$. The bottom is $(n - r)!$.
• Dividing by $p$ for a repeated letter. Divide by $p!$.
• Forgetting the orders inside a block. Different objects in a block can swap.
• A rank one too small. Add $1$ to the words before it.

Check yourself

1. Find ${}^9P_2$.

2. How many arrangements does the word ADDED have?

3. Four friends and a teacher stand in a row, and the teacher must stand at one end. How many rows are possible?

4. How many $2$ digit numbers have two different digits, each from $1$ to $9$?

5. In dictionary order, which arrangement of D, O, G comes first, and which comes last?