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# Permutation in mathematics made easy.Examples ,solutions,tricks and techniques.

## PERMUTATION

** **

· Permutation is an arrangement of objects considering its order.

If one event occurs in ‘m’ different ways and another event occurs independently in ‘n’ ways then the two events together can be done in (m x n) different ways.

If Mr. Maxon has 4 caps ,3 shirts and 2 paints then the number of ways he can dress is 4 x 3 x 2 that is 24 ways

**Fundamental counting Principle:** If the first task is done in “m” ways 2^{nd} task in “n” ways and the third in “r” ways. The total no. of ways all three task can be done is “m x n x r” ways.

· **Permutations:** Examples

a) Number formation using given digits.

b) Arrangement of books in a shelf.

c) Arrangement of people for a photograph etc

Permutation means arrangement of objects in which its order is very important.

· **Meaning of factorial:**

5! = 5 = 5 x 4 x 3 x 2 x 1 = 120

3! = 3 x 2 x 1 = 6

n! = n(n-1) (n-2) ………………… 3 x 2 x 1

· Meaning of ^{n}P_{r}→ Permutations of ‘n’ things taken ‘r’ at a time.

· **Formulae**

^{n}P_{r} = , ^{n}P_{n }=n! _{, }^{n}P_{1} = n, ^{n}P_{0} = 1, ^{n}P_{2} = n(n-1), 0! = 1

No. of diagonals in a polygon of ‘n’ sides = ^{n}C_{2} – n =

^{n}P_{r} means arrangement of ‘r’ number of objects from ‘n’ number of objects taken at a time.

Find the number of permutations of the letters of the word SECTION. Or How many different wasys the letters of the word SECTION can be arranged without repeating the letters? Like SECTION, ECTIONS, CTIONSE, SEIONCT etc. here all letters are taken at a time. Therefore it is ^{7}P_{7} =7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 ways.

The number of ways in which 3 books can be arranged in taking 2 at a time is.^{ 3}P_{2}

The number of 4 digit number can be formed using 1,2,3,4,5,6 is ^{6}P_{4}

The number of ways in which 4 students can be seated in a bench out of 5 students is ^{c}

How to find the value of ^{6}P_{4,}^{ 5}P_{4}, ^{3}P_{2} etc? Use this formula.^{ n}P_{r} = Here ‘n’ means total nember of objects and ‘r’ means the number of objects taken at a time and n!= n x (n-1)x(n-2)……….4 x 3 x 2 x 1.

(n-r) ! = (n-r) (n-r-1) (n-r-2) ………………4 x 3 x 2 x 1.

So ^{3}P_{2}= 3 x 2 x 1 / 2 x 1 = 3

^{6}P_{4 }= 6 x 5 x 4 x 3 x 2 x 1 / 4 x 3 x 2 x 1

Try these, Find how many 3 digit numbers can be formed using the numerals 6, 1, 2, 3, 4 without any repetitions?

Find the number of two digit numbers that can be formed using 4, 5, 6, and 8 without any repetitions?

Find the number of ways in which 4 students can be seated in a bench?

How many 4 digit even number can be formed using 1,2,3,4,8,9?

The number of ways we can arrange one red marble and one green marble always together in a shelf among another blue marble and a yellow marble is_________

The value of ^{20}C_{18} is________________

A, B, C and D are randomly called upon to participate in a debate. In how many ways B speaks immediately after A ?

In how many ways can 5 boys and 3 girls be seated in a row so that each girl is between two boys?

1. The value of ^{n}P_{r }when r=n is

a. 1 b. n c. n! d. r

2. If ^{6}P_{r} = 360 and ^{6}C_{r} = 15, then the value of ‘r’ is

a) 24 b) 4 c) 15 d) 36

3. If ^{11}P_{r}= 990, then ‘r’ is

a) 3 b) 4 c) 2 d) 5

4. If ^{n}C_{9} = ^{n}C_{6} , then ‘n’ is

a) 3 b) 15 c)10 d)14

5. Number of triangles can be formed by using 10 non collinear points

a) 100 b) 110 c) 120 d) 140

6. ^{n}P_{1} + ^{n}C_{1 }=

a) 2n b) n c) 2 d) n-1

7. Value of ^{20}C_{18 }is equal to

a) 360 b) 380 c) 190 d) 180

8. If ^{n}P_{3} = 120 , then ‘n’ is equal to

a) 6 b) 10 c) 8 d) 12

9. The value of 2! – 0! + 3! is

a)4 b) 5 c) 7 d) 8

10. The factorial form of 4 x 5 x 6 x 7 is

a) 7 b) 7! -3! c) d)

. Find the total number of 4 digit even numbers that can be formed using the numerals 0, 1, 2, 3, 5, 6 without any repetition.

31. Find the number of permutations of the letters of the word RICKET

32. There are 3 routes from P to Q. In how many ways one go from P to Q. In how many ways one go from P to Q

and return to P, if he does not like to return by the same route.

33. If ^{n}P_{r } = 60 and ^{n}C_{r} = 10 Find the value of r.

34. If ^{n}P_{4 }=12^{n}P_{2} then find ‘n’.

35. How many 3-digit numbers can be formed using the digits 1, 2, 3, 5 and 6 without repetitious? How many of

these are even numbers?

36. Find the total number of 2 digit numbers that to can be formed.

37. How many three digit numbers can be formed using the digits 0, 1, 2, 3 without repetitious?

38. In how many ways can 6 different books be arranged in a shelf o that 2 particular books are always together?

39. There are 8 members in a club of which A and B are two members. A committee of 5 be formed. How many ways of these will include A and exclude B?

Remember all these information is only for linear arrangement, here elements are not suppose to be repeat.. I f you want to learn other interesting concepts in easy way click here.