Very Short Answer on Permutations and Combinations Class 11 OP Malhotra Exe-12F ISC Maths Solutions Ch-12. In this article you would learn to solve all types questions on Permutations and Combinations. Step by step solutions of latest textbook has been given as latest syllabus. Visit official Website CISCE for detail information about ISC Board Class-11.
Permutations and Combinations Class 11 OP Malhotra Very Short Answer ISC Math Solutions Ch-12
| Board | ICSE |
| Publications | S Chand |
| Subject | Maths |
| Class | 11th |
| Chapter-12 | Permutations and Combinations |
| Writer | OP Malhotra |
| Exe-12(F) | Very Short Answer Type Questions. |
Very Short Answer on Permutations and Combinations
OP Malhotra ISC Class 11 Maths Solutions
Evaluate :
Que-1: P (7, 3)
Sol: P(n,r) = n! / (n-r)!
P(7,3) = 7! / (7-3)!
= 7! / 4!
= 7 × 6 × 5 = 210
Que-2: 8! / (6! × 2!)
Sol: (8 × 7 × 6!) / (6! × 2 × 1)
= (8 × 7) / 2
= 28
Que-3: If 4P2 = n × 4C2, find the value of n.
Sol: 4P2 = 4! / 2! = 4 × 3 = 12
4C2 = 4! / (2!2!) = 6
Given: 12 = n × 6
n = 2
Que-4: (i) How many 3 digit numbers can be formed by using the digits 1 to 9, if no digit is repeated?
(ii) How many 4 digit numbers can be formed by using the digits 0 to 9, if no digit is repeated?
Sol: (i) First digit → 9 choices
Second digit → 8 choices
Third digit → 7 choices
Total numbers = 9 × 8 × 7 = 504
(ii) First digit cannot be 0 → 9 choices
Remaining digits → 9 × 8 × 7
Total numbers = 9 × 9 × 8 × 7 = 4536
Que-5: In how many ways can 8 persons sit along a round table?
Sol: Circular permutation = (n − 1)!
= 7! = 5040
Que-6: In how many ways can the letters of the word RIVER be arranged?
Sol: Total letters = 5
R repeats twice
Number of arrangements = 5! / 2!
= 120 / 2
= 60
Que-7: Evaluate: 50C49 × 3P3
Sol: 50C49 = 50
3P3 = 3! = 6
Result = 50 × 6 = 300
Que-8: (i) Find the number of six-digit numbers all digits of which are odd.
(ii) How many 3 digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Sol: (i) Odd digits = 1,3,5,7,9 → 5 choices
Each position → 5 choices
Total numbers = 56 = 15625
(ii) Unit digit (even) → 2,4,6 → 3 choices
Hundreds digit → 6 choices
Tens digit → 6 choices
Total numbers = 6 × 6 × 3 = 108
Que-9: From a committee of 8 persons, in how many ways can a Chairman and a Vice-Chairman be chosen assuming one person cannot hold more than one position?
Sol: Chairman → 8 choices
Vice-Chairman → 7 choices
Total ways = 8 × 7 = 56
Que-10: In a 12 storeyed building, 3 persons enter a lift cabin. It is known that they will leave the lift at different floors. In how many ways can they do so if the lift does not stop at the second floor?
Sol: Possible floors = 11 (excluding ground)
Since lift does not stop at 2nd floor → remaining floors = 10
3 persons leave at different floors → permutations =
10P3 = 10 × 9 × 8 = 720
Que-11: In a football championship, 153 matches were played. Every two teams played one match with each other. Find the number of teams participating in the championship
Sol: Number of matches = n(n−1)/2
n(n−1)/2 = 153
n(n−1) = 306
n = 18 teams
Que-12: If (1/8!) + (1/9!) = x/10! , find x.
Sol: 1/8! + 1/9!
= 9/9! + 1/9!
= 10/9!
= 100/10!
Therefore x = 100
Que-13: How many 6-digit telephone numbers can be constructed with the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Sol: First two digits fixed → 67
Remaining digits = 8 digits left
Choose 4 places without repetition
= 8P4 = 8 × 7 × 6 × 5 = 1680
Que-14: There are 5 red, 4 white and 3 blue marbles in a bag. They are drawn one by one and arranged in a row. Assuming that all the 12 marbles are drawn, determine the number of different arrangements.
Sol: Total marbles = 12
Arrangements = 12! / (5! 4! 3!)
= 27720
Que-15: 18 mice were placed in two experimental groups and one control group with all groups equally large. In how many ways can the mice be placed into three groups?
Sol: Each group = 6 mice
Ways = 18! / (6! 6! 6!)
= 17153136
Que-16: Find the number of ways in which 12 apples may be equally divided among 3 children.
Sol: Total apples = 12
Total children = 3
Number of ways to distribute 12 apples among 3 children
using combinations is given by:
Number of ways = C(12 + 3 − 1, 3 − 1)
= C(14, 2)
= (14 × 13) / 2
= 34,650
Que-17: (i) A man has 6 friends. In how many ways may he invite one or more of them to dinner?
(ii) Similar question. There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
Sol: (i) Each friend may be invited or not invited.
Total ways = 26 − 1 = 63 ways
(ii) Each lamp has two choices (ON or OFF).
Total ways = 210 − 1 = 1023 ways
Que-18: In how many ways can the results of three successive football matches be decided?
Sol: Each match has 3 results: Win, Lose, Draw.
Total ways = 3 × 3 × 3 = 33 = 27 ways
Que-19: How many automobile license plates can be made, if each plate contains two different letters followed by three different digits?
Sol: Letters: 26P2 = 26 × 25
Digits: 10P3 = 10 × 9 × 8
Total plates = (26 × 25)(10 × 9 × 8)
= 468000 plates
Que-20: A box contains two white, three black and four red balls.
In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
Sol: Total balls = 9
Total ways = 9C3 = 84
Without black balls = 6C3 = 20
Required ways = 84 − 20 = 64 ways
Que-21: (i) Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no two vowels are together.
(ii) Similar question. Find the number of ways in which 5 girls and 3 boys can be seated in a row so that no two boys are together.
Sol: (i) Letters = 8 (3 vowels, 5 consonants)
Arrange consonants = 5! = 120
Spaces = 6
Choose spaces for vowels = 6C3 = 20
Arrange vowels = 3! = 6
Total ways = 120 × 20 × 6 = 14400 words
(ii) Arrange girls = 5! = 120
Spaces = 6
Choose spaces for boys = 6C3 = 20
Arrange boys = 3! = 6
Total ways = 120 × 20 × 6 = 14400 ways
Que-22: If 56Pr+6 : 54Pr+3 = 30800 : 1, then r = ______.
Sol: Using permutation formula:
nPr = n!/(n−r)!
After simplification:
r = 4
–: End Permutations and Combinations Class 11 OP Malhotra Exe-12F ISC Maths Ch-12 Solutions :–
Return to :- OP Malhotra ISC Class-11 S Chand Publication Maths Solutions
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