PERMUTATIONS: The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or collection of objects with due regard being paid to the order of arrangement or selection, are called permutations.
COMBINATIONS: The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important, are called combinations.
Results
#1. ⁴P₃ is evaluated as
⁴P₃ is evaluated as
4= 4*3(3-2 1)
=12(2)
Correct answer is =24
#2. The number of ways the letters of the word COMPUTER can be rearranged is
The number of ways the letters of the word COMPUTER can be rearranged is
COMPUTER = 8
8!= 8*7*6*5*4*3*2*1
number of ways the letters of the word COMPUTER can be rearranged is = 40320
#3. The sum of all 4 digit number containing the digits 2, 4, 6, 8 without repetition is
The sum of all 4 digit number containing the digits 2, 4, 6, 8 without repetition is
4!= 4 * 3 * 2 * 1
=24
how many time each digit= 24/4
= 6
sum of at 1000 digits =6(2 4 6 8 )1000
= 120000
sum of 100 digits =6(2 4 6 8 )100
=12000
sum of 10 digits = 6(2 4 6 8 )10
= 1200
sum of unit digits = 6(2 4 6 8 )1
120
sum of all numbers= 120000 12000 1200 120
=133320
#4. If ⁵Pʳ = 60, then the value of r is
If ⁵Pʳ = 60, then the value of r is
by using formula =nPr= n!/(n-r)!
5!= 5*4*3*2*1
=120
60 = 120/(5-r)!
120/60=(5-r)!
2!=(5-r)!
2=5-r
2-5=-r
-3=-r
r=3
#5. The number of ways in which 7 girls form a ring is
The number of ways in which 7 girls form a ring is
7!(7-1)=6
6*5*4*3*2*1=720
#6. The number of numbers lying between 100 and 1000 can be formed with the digits 1, 2, 3,4, 5, 6, 7 is
The number of numbers lying between 100 and 1000 can be formed with the digits 1, 2, 3,4, 5, 6, 7 is
100 and 1000 = 3 digits numbers
total digits = 7
7P3
7!=7*6*5
=210
210 number s lying between 100 and 1000
#7. The total number of 9 digits numbers of different digits is
The total number of 9 digits numbers of different digits is
1, 2 , 3, 4, 5, 6,7 ,8,9
9*9!
#8. 0! is a symbol equal to
0! is a symbol equal to 1
#9. The number of ways in which the letters of the word DOGMATIC can be arranged is
The number of ways in which the letters of the word DOGMATIC can be arranged is
DOGMATIC total digits = 8
8!= 8*7*6*5*4*3*2*1
=40320
#10. How many four digits number can be formed by using 12...............7?
How many four digits number can be formed by using 12……………7?
total number =7
and 4 digits
7P4
#11. A person has 8 friends. The number of ways in which he may invite one or more of them to a dinner is.
A person has 8 friends. The number of ways in which he may invite one or more of them to a dinner is.
(2)8=256-1
255
#12. There are 12 points in a plane of which 5 are collinear. The number of triangles is
There are 12 points in a plane of which 5 are collinear. The number of triangles is
we will Divide the Triangle into 3 categories
Triangle with all the vertices from (12-5 = 7) points
Number of Triangle = ⁵C₂*⁷C₁ ⁵C₁*⁷C₂ ⁷C₃
= 10*7 5*21 35
= 70 105 35
= 210
#13. The number of different factors the number 75600 has is
The number of different factors the number 75600 has is Total number of factors
75600 factors are 2*2*2*2*3*3*3*5*5*7
2*4,3*3,5*2,7*1
(4 1)(3 1)(2 1)(1 1)=120
There are 120 factors of the number 75600.
#14. If ⁵⁰⁰C₉₂= ⁴⁹⁹C₉₂ + nC₉₁ then x is
If ⁵⁰⁰C₉₂= ⁴⁹⁹C₉₂ nC₉₁ then x is
x value is =499
#15. If ¹⁰Pr= 604800 and ¹⁰Cr=120; find the value r
¹⁰Pr =¹⁰Cr.r!
604800=120*r!
r!=604800/120=5040
r!=5040=7*6*4*3*2*1=7!
r=7
#16. In how many ways can 9 letters be posted in 4 letter boxes?
9 letters be posted in 4 letter boxes
there are 4⁹ ways
#17. In how many ways can 8 beads of different color be strung on a ring
In how many ways can 8 beads of different color be strung on a ring
there 7!/2 ways
=2520
#18. How many telephone connection may be allotted with 8 digits form the numbers 012......9?
How many telephone connection may be allotted with 8 digits form the numbers 012……9?
0 ,1,2,3,4,5,6,7,8,9
10 digits
numbers of combinations are = 10⁸ answer
#19. In how many ways can 4 boys and 3 girls stand in a row so that no two girls are together?
In how many ways can 4 boys and 3 girls stand in a row so that no two girls are together?
correct answer is =5! * 4! /3!
#20. How many digits numbers greater than 7000 can be formed out of the digits 3,5,7,8,9?
or the number to be greater than 7000 , first place can be filled with anyone of 7,8,9.Thus first place can be filled in 3 ways.
The next places can be filled with the 4,3,2 ways
Thus total number of ways= 3*4*3*2=72 ways