Permutations Test
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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