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Permutation and Combination
⚪ What is Factorial ?⚫ Let n be a positive integer. Then , the continued product of first n natural numbers is called factorial n , to be denoted by n!
n! = n(n-1)(n-2)........3.2.1
⚪ What is Permutation ?
⚫ The ways of arranging or selecting a smaller or an equal number of persons or objects at a time from a given group of persons or objects with due regard being paid to the order of arrangement or selection are called permutation
for example: Three different things a, b and c are given , then different which can be made by taking two things from three given things are ab , ac ,bc, ba , ca , cb
Therefore, the number of permutation is 6
⚪ What is Combination ?
⚫ Each of the different groups or selections which can be formed by taking somne or all of a number of objects , irrespective of their arrangements , is called combination.
The number of all combinations on n things , taken r at a time is denoted by C(n, r) or nCr
Some Important Rules :
1. Permutation of 'n' different things taken at a time (all) then P of them are alike of one kind , when Q are like of second kind , R are alike of third kind are given by :
Example : In how many ways we can arrange 'Mathematics'
A occurs 2 times , T occurs 2 times , M occurs 2 times
No. of arrangement =
2. The number of ways of selecting 0 or more things from n different things is given by
nC0 + nC1 + ...... + nCn = 2n
3. The number of selection of one or more things from p + q+ r + s in which p is alike , q is alike 2nd kind , r is alike of 3rd kind , s is different are (p+1)(q+1)(r+1)2s-1
For example : AABBBCCCDDEF
A = 2 =p
B = 4 = r
C = 3 = q
d = 2 = s
E, F = 2
Number of selection = (2+1)(4+1)(3+1)(2+1)22 - 1
3. The number of ways in which n person and n things can be used to form a ring :
(i) (n-1)! [if clockwise or anticlockwise not required]
(ii) (n-1)!/2 [when clockwise or anticlockwise required]
5. If 'm' is the index of the highest power of a prime number p which divides the factoral n , then
Example :
5. If 'n' straight line are drawn in a plane such that no two lines are parallel , and no three are concurrent , then the no. of parts which these lines divides the plane , then the no. of parts which these lines divides the plane is given by
Example : No. of parts = 1 + n(n+1)/2 = 1 + 3*4/2 = 7
6. The number of diagonal of n side polygon :
No. of diagonal = nC2 - n
7. The Number of straight lines that can be made by joining n non collinear point Number of straight line = nC2 and if p point are collinear , then number of straight lines Number of straight line = nC2 - pC2 + 1
8. The number of triangle that can be made by n non collinear Number of triangle = nC3 and if p point are collinear , then number of triangle Number of triangle = nC3 - pC3 + 1
9. Maximum number of points of intersection of n circle : Number of points = nP2
Example :
Number of maximum points = nP2 = 3P2 = 6
10. Maximum number of points of intersection of n straight lines Number of points = nC2
11. The number of ways in which m (one type of different things) and n (another type of different things) can be arranged in a row such that all the second types things come together No. of ways = n!(m+1)!
Example : 5 Boys and 3 girls can seat in arow such that all the girls together = 3 !(5+1)! = 3!.6!
11. The number of ways in which n (one type of different things) and n-1 (another type of different things) can be arranged in a row such that no two things come together No. of ways = n!(n-1)!
Example : 5 Boys and 4 girls can seat in a row such that no boy and girl seat together = 5 !(5-1)! = 5!4!
12. The number of ways in which n (one type of different things) and n (another type of different things) can be arranged in a row alternatively No. of ways = 2(n!n!)
Example : 5 Boys and 5 girls can seat in a row alternatively = 2.5!5!
Practice set 1 Practice set 2 Practice set 3
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