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.
When the elements of a set are arranged in a definite order, the arrangement is called a permutation of the elements. The number of permutations of n objects is n!
The number of possible orderings of m objects taken from a set of n is given by:
Pnm=n×(n−1)×(n−2)×⋯×(n−m+1) =n!(n−m)!
That is, count backwards starting from n , writing down the numbers as you count, until you've written down m numbers. Then multiply them all together.
Example:
Suppose you're a television programmer, and you have five half-hour shows to choose from, but only three time slots. How many different programs are possible?
Using the permutations formula, we have:
P53=5×4×3=60
To see why this works, name the shows A, B, C, D, and E, and make a list:
ABC
ABD
ABE
ACB
ACD
ACE
ADB
ADC
ADE
AEB
AEC
AED
BAC
BAD
BAE
BCA
BCD
BCE
BDA
BDC
BDE
BAC
BAD
BAE
CAB
CAD
CAE
CBA
CBD
CBE
CDA
CDB
CDE
CEA
CEB
CED
DAB
DAC
DAE
DBA
DBC
DBE
DCA
DCB
DCE
DEA
DEB
DEC
EAB
EAC
EAD
EBA
EBC
EBD
ECA
ECB
ECD
EDA
EDB
EDC
In this case, there are 5 choices for the first program, 4 choices for the second program, and 3 choices for the last program. So the answer is:
answered
__________1. What do you call the selection and arrangement of objects in a particular order? a. permutation b. combination
c. probability d. statistics
__________2. Two different arrangements of objects where some of them are identical are called .
a. Distinguishable permutations
c. circular permutations
b. Unique combinations
d. circular combinations
__________3. The manner of solving by multiplying the number of outcomes or objects of two or more events to find the total number of outcomes for those events to occur is known as _______
a. Permutation b. combination
c. Fundamental Counting Principle d. factorial
__________4. What do you call the different possible arrangements of objects in a circle?
a. Distinguishable permutations
c. circular permutations
b. Unique combinations
d. circular combinations
__________5. It is the process of multiplying consecutive decreasing whole numbers from an identified number down to one.
a. Permutation b. combination
c. FCP d. factorial notation
__________6. What do you call the selection of objects in which order does not matter?
a. permutation b. combination
c. probability d. statistics
__________7. Which of the following situations or activities involve permutation?
a. Matching shirts and pants
b. Forming different triangles out of 5 points on a plane, no three of which are collinear
c. Assigning telephone numbers to subscribers
d. Forming a committee from the members of a club.
1
1 A
I mean 1. A
2. C
3. C