This ar covers permutations and also combinations.

You are watching: How many ways can the letters be arranged

**Arranging Objects**

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

**Example**

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

The prize is 4! = 24.

This is since there are four spaces to it is in filled: _, _, _, _

The an initial space can be fill by any one the the 4 letters. The 2nd space deserve to be fill by any kind of of the remaining 3 letters. The third an are can it is in filled by any type of of the 2 remaining letters and also the final an are must be filled through the one continuing to be letter. The total number of possible arrangements is thus 4 × 3 × 2 × 1 = 4!

The number of ways of arranging n objects, the which ns of one kind are alike, q that a second form are alike, r of a third kind are alike, etc is:

n! .p! q! r! …

**Example**

In how many ways can the letters in the word: STATISTICS be arranged?

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

10!=50 4003! 2! 3!

**Rings and also 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 many different ways have the right to they be seated?

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

**Combinations**

The number of ways of picking r objects native n unlike objects is:

**Example**

There room 10 balls in a bag numbered from 1 to 10. Three balls space selected at random. How many different means are there of selecting the 3 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 actually to pick the height 3 goals out that 10. Since the bespeak is important, the is the permutation formula which we use.

10P3 =10! 7!

= 720

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

**Probability**

The over facts deserve to be supplied to help solve troubles in probability.

**Example**

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

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

See more: What Does X In Lax Stand For, What Does The X In Lax Stand For

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