This ar covers permutations and also combinations.

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Arranging Objects

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


How countless different ways have the right to the letters P, Q, R, S be arranged?

The prize is 4! = 24.

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

The an initial space can be filled by any kind of one of the 4 letters. The second space can be to fill by any of the continuing to be 3 letters. The third room can it is in filled by any type of of the 2 staying letters and the final an are must it is in filled by the one continuing to be letter. The total variety of possible kinds is as such 4 × 3 × 2 × 1 = 4!

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

n! .p! q! r! …


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

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

10!=50 4003! 2! 3!

Rings and also Roundabouts

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

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


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

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


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



There space 10 balls in a bag numbered indigenous 1 come 10. Three balls room selected at random. How numerous different ways are there of selecting the 3 balls?

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


A permutation is an notified arrangement.

The variety of ordered arrangements of r objects taken from n uneven objects is:

nPr = n! . (n – r)!


In the match of the Day’s goal of the month competition, you had actually to choose the top 3 objectives out that 10. Since the stimulate is important, the is the permutation formula which we use.

10P3 =10! 7!

= 720

There are therefore 720 different ways of picking the top three goals.


The over facts deserve to be used to aid solve troubles in probability.


In the national Lottery, 6 numbers are chosen from 49. You success if the 6 balls girlfriend pick enhance the six balls selected through the machine. What is the probability of win the nationwide Lottery?

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

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