This section covers permutations and also combinations.

You are watching: How many arrangements of the letters in the word "schools" are there?

Arranging Objects

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


How many different ways deserve to the letters P, Q, R, S it is in arranged?

The prize is 4! = 24.

This is since there are 4 spaces to be filled: _, _, _, _

The very first space deserve to be to fill by any one of the 4 letters. The 2nd space have the right to be fill by any kind of of the remaining 3 letters. The third room can it is in filled by any of the 2 staying letters and also the final room must be filled by 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, of which p of one type are alike, q of a second form are alike, r of a third type are alike, and so on is:

n! .p! q! r! …


In how numerous ways have the right to the letter in the word: STATISTICS be arranged?

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

10!=50 4003! 2! 3!

Rings and Roundabouts

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

When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!


Ten civilization go to a party. How numerous different ways have the right to they be seated?

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


The variety of ways of choosing r objects indigenous n uneven objects is:



There are 10 balls in a bag numbered from 1 to 10. Three balls space selected at random. How plenty of different ways are there of picking the three balls?

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


A permutation is an ordered arrangement.

The number of ordered arrangements of r objects taken from n unlike objects is:

nPr = n! . (n – r)!


In the complement of the Day’s score of the month competition, you had to choose the top 3 purposes out of 10. Because the order 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.


The above facts can be provided to assist solve troubles in probability.


In the nationwide Lottery, 6 number are preferred from 49. You win if the 6 balls girlfriend pick match the six balls selected by the machine. What is the probability of to win the nationwide Lottery?

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

See more: How Many Staal Brothers Are There, Jared Staal

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