*

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

The variety of ways that arranging n unequal objects in a ring when clockwise and also anticlockwise arrangements are various is (n – 1)!

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.