Permutations Test

Introduction to Permutation test:

Permutation means arrangement of thing. The operation of arranging the order of objects in particular order, as ‘pqrs’ into ‘rpqs', ‘qspr', etc., or of arranging a number of objects in groups made up of equal numbers of the objects in different orders, as 'x' and 'y' in 'xy' and 'yx' ;  A one-to-one transformation of a set with a finite number of objects.

Permutation Formula Used in Tests:

If k is the number of possible permutations from a set of n then this is usually written nPk .


Formula: nPk = `(n!)/((n-k)!) "= n(n - 1)(n - 2)..........(n - k + 1)"`

Proof:     Let us consider we have ‘n’ different things a1, a2……, an.

First place can be filled up in ‘n’ different ways.

Number of ways to filled up things left after first place = n-1

So the second-place can be filled-up in (n-1) ways.

Now number of ways to  filling-up things left after the first and second places = n - 2

Now the third place can be filled-up in (n-2) ways.

Thus number of ways of filling-up first-place = n

Number of ways of filling-up second-place = n-1

Number of ways of filling-up third-place = n-2

Number of ways of filling-up k-th place = n – (k -1) = n-k+1

By multiplication – rule of counting,

Total no. of ways of filling up, first, second ..... up to  kth-place together :-

n (n-1) (n-2) ------------ (n-k+1)

Hence:
nPr       = n (n-1)(n-2) --------------(n- k+1)

= `("[n(n - 1))(n - 2)................(n - k +1)][(n - k)(n - k -1)..........3. 2. 1]")/("[(n - k)(n - k- 1)].......3. 2. 1")`

nPk = `(n!)/((n - k)!)`

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Test Problems in Permutation:

Answer the following permutation test problems:

1)How many 4-digit numbers can be arranged from the digits 2, 5, 6, 9, and 7, if each digit is distinct ?

2)How many 7-letters can be arranged from the letters ‘a-z’, if each letter is distinct?

3)Find the number of the ways to select and rank favorite 6 days of the leap year.

4)How many ways can 3 students from group of 12 are lined up for a photograph?

Problem 1:How many 4-digit numbers can be arranged from the digits 2, 5, 6, 9, and 7, if each digit is distinct?

Solution: Counting the number of permutations ‘P’ of 5 distinct elements, taken 5 at a time. The number of permutations ‘P’ of ‘n’‘r’ at a time is unique elements, taken

nPr = `(n!)/((n-k)!)`

5P4 = `(5!)/((5-4)!)` = `(5!)/(1)!`5! / 1! = 5x4x3x2 = 120

Thus, 120 different 4-digit numbers can be formed from the digits 2, 5, 6, 9, and 7.

Problem 2:How many 7-letters can be arranged from the letters ‘a-z’, if each letter is distinct?

Solution:Counting the number of permutations ‘P’ of 26 distinct letters, taken 3 at a time. The number of permutations ‘P’ of ’n’‘r’ at a time is distinct objects, taken

nPr =`(n!)/((n-k)!)`

n = 26, r = 7

26P7 = `(26!)/((26-7)!)` = `(26!)/(19)!` = 26x25x24x23x22x21x20 = 3315312000.

Thus, 3315312000 different 7-letters can be formed from the letters ‘a-z’.

Problem 3:Find the number of the ways to select and rank favorite 6 days of the leap year.

Solution:n = 366, r = 6

nPr = `(n!)/((n-k)!)`

366P6 = `(366!)/((366-6)!)` = `(366!)/(360!)` = 366x365x364x363x362x361= 2306735136866160

Thus, 2306735136866160 different 6-days can be formed from the leap year.

Problem 4:How many ways can 3 students from group of 12 are lined up for a photograph?

Solution: Choosing  3 students  from 12  and arranging them is

12P3     =      12x11x10 = 1320

Test Problems in Permutation for Practice:

Problem 1:How many ways can 4 students from group of 15 are lined up for a photograph?

Answer: 32760 

Problem 2:How many 4-digit numbers can be arranged from the digits 3, 4, 5, 6, 7, 8, and 9, if each digit is unique?

Answer: 840