This ar covers permutations and also combinations.

You are watching: How many ways are there to rearrange the letters in function?

**Arranging Objects**

The variety of ways of arranging n unlike objects in a heat is n! (pronounced ‘n factorial’). N! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

**Example**

How many different ways have the right to the letter P, Q, R, S be arranged?

The price is 4! = 24.

This is due to the fact that there are 4 spaces to be filled: _, _, _, _

The first space can be filled by any one the the four letters. The second space can be to fill by any type of of the staying 3 letters. The third an are can be filled by any type of of the 2 staying letters and also the final space must it is in filled through the one remaining letter. The total variety of possible arrangements is thus 4 × 3 × 2 × 1 = 4!

The variety of ways that arranging n objects, of which p of one kind are alike, q that a second type are alike, r of a third kind are alike, and so on is:

n! .p! q! r! …

**Example**

In how countless ways have the right to the letters in the word: STATISTICS be arranged?

There are 3 S’s, 2 I’s and also 3 T’s in this word, therefore, the variety of ways of arranging the letter are:

10!=50 4003! 2! 3!

**Rings and Roundabouts**

When clockwise and also anti-clockwise arrangements space the same, the variety of ways is ½ (n – 1)!

**Example**

Ten world go to a party. How countless different ways deserve to they be seated?

Anti-clockwise and also clockwise arrangements are the same. Therefore, the total variety of ways is ½ (10-1)! = 181 440

**Combinations**

The number of ways of selecting r objects indigenous n unequal objects is:

**Example**

There space 10 balls in a bag numbered from 1 come 10. 3 balls space selected in ~ random. How plenty of different ways are there of choosing the three balls?

10C3 =10!=10 × 9 × 8= 120 3! (10 – 3)!3 × 2 × 1

**Permutations**

A permutation is an notified arrangement.

The number of ordered species of r objects taken from n unequal objects is:

nPr = n! . (n – r)!

**Example**

In the enhance of the Day’s score of the month competition, you had to choose the peak 3 goals out the 10. Due to the fact that the order is important, the is the permutation formula which we use.

10P3 =10! 7!

= 720

There are therefore 720 various ways of choose the height three goals.

**Probability**

The over facts deserve to be used to help solve problems in probability.

**Example**

In the nationwide Lottery, 6 number are liked from 49. You win if the 6 balls you pick complement the six balls selected through the machine. What is the probability of winning the nationwide Lottery?

The variety of ways of selecting 6 numbers from 49 is 49C6 = 13 983 816 .

See more: 2008 Dodge Caliber Serpentine Belt Diagram For 2008 Dodge Caliber

Therefore the probability of win the lottery is 1/13983816 = 0.000 000 071 5 (3sf), i beg your pardon is about a 1 in 14 million chance.