This section covers permutations and combinations.
Arranging Objects
The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1
Example
How many different ways can the letters P, Q, R, S be arranged?
The answer is 4! = 24.
This is because there are four spaces to be filled: _, _, _, _
The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 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 type are alike, r of a third type 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 number of ways of arranging the letters are:
10!=50 400
3! 2! 3!
Rings and Roundabouts
- The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)!
When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!
Example
Ten people go to a party. How many different ways can they be seated?
Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440
Combinations
The number of ways of selecting r objects from n unlike objects is:
Example
There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls?
10C3 =10!=10 × 9 × 8= 120
3! (10 – 3)!3 × 2 × 1
Permutations
A permutation is an ordered arrangement.
The number of ordered arrangements of r objects taken from n unlike objects is:
nPr = n! .
(n – r)!
Example
In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.
10P3 =10!
7!
= 720
There are therefore 720 different ways of picking the top three goals.
Probability
The above facts can be used to help solve problems in probability.
Example
In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?
The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 .
Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.
Answer by robertb(5830) You can put this solution on YOUR
website! |
Continue Learning about Other Math
How could you find the number of different combinations of six letters in a computer password?
When trying to work out how many different combinations there are, you need to know how many options there are for each value. If the password only contains lower case letters, then we have 26 options for each value. For each letter in the password, there are 26 options, so the total number of possible options is 26x26x26x26x26x26 or 266 This equals 308,915,776 so there are 308,915,776 possible different combinations of six letters.
How many different 2 number combinations are with 243?
Three combinations: 23, 24 and 34
A keycode must contain 2 letters and 3 numbers. The letters may be any letter of the alphabet. The numbers should be any number from 0 to 9. How many different keycode combinations are there?
676,000 but I don’t know why
How many possible combinations of three letters are there?
Say you have the letters A,B, and C. Here are all the possible combinations. * ABC * ACB * BAC * BCA * CAB * CBA So, 6 if you don't repeat any of the letters. If you DO repeat letters, then simply take the number of letters you have, (3 for instance), and multiply it to the power of the number of letters you have. So, for 3 letters, the formula would be 33 . Or if you had 4 letters it would be 44 and so on.
If using 7 single digit numbers how many different combinations can you get?
Number of 7 digit combinations out of the 10 one-digit numbers = 120.