A permutation is an act of arranging numbers or objects in order.
For example, if we arrange the letters C, A and P, we get 6 different permutations as follows:
CAP ACP PAC
CPA APC PCA
Note that the position of letters is important in a permutation.
The way of selecting the objects from a group of objects is called combinations.
For example, if we wish to choose 2 letters from the group C, A and P, we get 3 combinations as follows:
CA CP AP
Here AC = CA because the position of these letters is not important in a combination.
Listing method:
Example: How many different four-digit numbers can be made by using the digits 2, 4, 6 and 8 assuming each digit can be used only once?
Solution:
List down all the three-digit numbers possible using the digits 2, 4 and 8
248 284 482
284 842 824
6 different three-digit numbers can be made. This is the case of permutation as the position of digits is important here.
Example: How many groups can be made from the word "MATHS" if each group consists of 3 alphabets?
Solution:
List down all the possible groups that can be made using the word ‘MATHS’
MAT MAH MAS HAT
AHS THS TSM HSM
HMT SAT
Total number of groups = 10
Here MAT = ATM = TAM = TMA as the position is not important here. This is a simple example of a combination.
There are three people A, B and C and two chairs. In how many ways these three people can occupy the two chairs?
Solution:
Let the boxes below represent two chairs.
List down all the possibilities.
The total number of possibilities = 6
This is the case of permutation as the position of people (A, B and C) is important here.
n! is the number of ways of arranging n unlike objects in a line.
n!(pronounced 'n factorial') = n × (n-1) × (n-2) × .... 3 × 2 × 1
Permutation formula:
The number of ways of arranging n objects, of which a of one type are alike, b of second type is alike, etc is:
n! ÷ a! b! .....
The number of ways of arranging n unlike objects in a circle or in a ring when clockwise and anticlockwise arrangements are different is (n-1)! When anti-clockwise and clockwise arrangements are the same, the number of ways is 1⁄2(n-1)!
Let us now learn how to apply these formulas using a few examples.
Example: How many six-digit numbers can be formed using the digits given below:
3, 5,6,6,7,8
Solution: Number of digits to be arranged i.e. n = 6
Number of times digit 3 is repeating i.e. a = 2
Using permutation formula,
Number of ways in which 6 digits can be arranged
= n! ÷ a!
= 6! ÷ 2! = 6 × 5 × 4 × 3 × 2 × 1 ÷ 2 × 1 = 6 × 5 × 4 × 3 = 360
360 six-digit numbers can be formed using the given digits.
Example: Work out the number of ways in which 6 women can be arranged at the round table.
Solution: Number of women to be arranged at the round table i.e. n = 6
Using permutation formula for arranging objects in a circle,
Number of ways = (n - 1)!
= (6 - 1)!
= 5!
= 5 × 4 × 3 × 2 × 1
=120
6 women can be arranged at the round table in 120 ways.
What counts as a "good" score will vary depending on the school you want to attend. The standardized 11 Plus test score average across the country is roughly 100. The highest average in some areas is 111. The lowest scores would often fall between 60 and 70, while the highest scores would normally fall between 130 and 140. To achieve excellent marks on 11+ Maths Exams, practice 11+ Maths topic-wise questions.
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