OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Pr = n!/(n-r)! ; 0 ≤ r ≤ n

Where the symbol “!” denotes the factorial which means that the product of all the integer less than or equal to n but it should be greater than or equal to 1.

For example,

1! = 1

2! = 1 x 2 =2

3! = 1 x 2 x 3 = 6

4! = 1 x 2 x 3 x 4 = 24, which are the factors of the given number.

Permutation When all the Objects are Distinct

There are some theorems involved in finding the permutations when all the objects are distinct. They are :

Theorem 1: If the number of permutations of n different objects taken r at a time, it will satisfy the condition 0 < r ≤ n and the objects which do not repeat is n ( n – 1) ( n – 2)……( n – r + 1), then the notation to denote the permutation is given by “ ⁿ P_r”

Theorem 2: The number of permutations of different objects “n” taken r at a time, where repetition is allowed and is given by n^r .

Permutation When all the Objects are not Distinct Objects

Theorem 3: To find the number of permutations of the objects ‘n’, and ‘p’ are of the objects of the same kind and rest is all different is given as n! / p!

Theorem 4: The number of permutations of n objects, where p₁ are the objects of one kind, p₂ are of the second kind, …, p_k is of the k^th kind and the rest, if any, are of a different kind then the permutation is given by n! / ( p₁!p₂!…P_k!)

Combination Definition

The combination is a selection of a part of a set of objects or selection of all objects when the order doesn’t matter. Therefore, the number of combinations of n objects taken r at a time and the combination formula is given by;

Relationship Between Permutation and combination

In permutation and combination for class 11, the relationship between the two concepts is given by two theorems. They are;

Theorem 5: ⁿP_r = ⁿC_r r! ; if 0 < r ≤ n.

Theorem 6: ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

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Exe-12 (a)

OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Page 12-5 to 12-6

Question 1:

Two person go in a railway carriage where there are  6 ……………….. they seat themselves ?

Question 2:

In how many ways …………………  get both the price.

Question 3:

…………………….

……………………

……………………

Question 18:

For a seat of five true ………………………… this to be possible ?

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Exe-12 (b)

OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Page 12-9

Question 1:

Evaluate :

4!/2! 2!

Question 2:

……………………

……………………

…………………..

Question 20:

Convert into factorial :

7, 8, 9, 10, 11, 12, 13, 14, 15

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Exe-12 (c)

OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Page 12-20 to 12-21

Question 1:

12 different books a self will hold five ……………………….. made on the self.

Question 2:

………………………..

………………………..

………………………..

Question 31:

In how many ways ………………………. a lady by his side ?

Question 32:

The letter  of the word ZENITH are …………………… rank of word ZENITH ?

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Exe-12 (d)

OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Page 12-26

Question 1:

Find the value of :

(i) …

(ii)……….

(ii)………..

Question 2:

……………………

…………………….

……………………..

Question 11:

​\( C_{3}^{x-1}\textrm{} + C_{4}^{x-1}\textrm{} > c_{3}^{x} \)​  of

(a) x > 7

(b) x ≥ 7

(c) x >6

(d) n ≥ 6

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Exe-12 (e)

OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Page 12-30 to 12-31

Question 1:

In how many ways …………………… from 10 individuals ?

Question 2:

In how many ways can a committee………………….. ……………….. taken everytime ?

Question 3:

………………………

………………………

………………………

Question 28:

Find the number of word which can be ………………………………… word “COMBINATION”.

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Chapter Test

OP Malhotra Class-11 Permutations and Combinations S.Chand ISC Maths Solutions

Page 12-34 to 12-35

Question 1:

Find the value of P (7, 3)

Question 2:

How many number of 5 digit ………………….. digit more than once.

Question 3:

…………………………

………………………..

………………………..

Question 10:

Find the number of ways in which 12 apples may be equally divided among 3 children.

 

-: End of Permutations and Combinations Solution :-

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