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subject: Permutations Test [print this page]

Introduction to Permutation 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)!)'

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 nr 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 nr 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

by: Omkar Nayak

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