1. Factorial notation-
The continuous product of first of n natural number is called factorial
i.e. Ðn or n! = 1,2,3,
..(n-1).n
n! =n(n-1)! = n(n-1)(n-2)! & so on
or n(n-1)
.. (n-r+1) = (n!/(n-r)!))
Here 0! = 1 and (-n)! = meaningless.
2. Fundamental principle of counting
(i) Addition rule: If there are two operation such that they can be done independently in m and n ways respectively, then either (any one) of these two Addition Þ OR (or) Option
(ii) Multiplication rule: Let there are two tasks of an operation and if these two tasks can be performed in m and n different number of ways respectively, then the Multiplication Þ and (or) Condition
(iii) Bijection Rule: Number of favourable cases = Total number of cases Unfavourable number of cases.
3. Permutations (Arrangement of obiects )-
(i) The number of permutations of n different things taken e at a time is npr = (n!/(n-r)!)
(ii) The number of permutations of n dissimilar things taken all at a time is npr = n!
(iii) the number of permutations of n distinct obects taken r at a time, when repetition of object is allowed is n.
(iv) If uot of n object, a are alike of one kind, b are alike of second kind and c are alike of thied kind and the rest distinct, then the number of wayes of permuting the n object is (n!/(a!b!c!))
4. Restricted permutations-
(i) The number of permutation of n dissimilar things taken r at a time, when m particular things always occupy definite places = n-mpr-m
(ii) The number of permutations of n dissimilar things taken r at a time, when m particular things are always to be excluded (included) = n-mpr(n-mCr-m x r!)
5. Circular permutation-
When clockwise & anticlockwise order are treated as different
(i) The number of circular permutations of n different things taken r at time (npr /r)
(ii) The number of circular permutations of n different things taken altogether (npn / n) = (n-1)!
When clockwise & anticlockwise orders are treated as same.
(i) The number of circular permutations of n different things taken all together (npr/2r)
(ii) The number of circular permutations of n different things taken all together (npn/2n) = (1/2) (n-1)!
6. Combination (selection of objects)-
The number of combination of combination of n different things taken r at a time is denoted by nCr or C (n,r)
(i) nCr = nCn-r
(ii) nCr + nCr-1 = n+1Cr
(iii) nCr = nCs Þ r =s or r+s = n
(iv) nC0 = nCn = 1
(v) nC1 = nCn-1 = n
(vi) nCr = n/r n-1Cr-1
(vii) nCr = (1/r) (n-r+1) nCr-1
7. Restricted combinations-
The number of combination of n distinct object taken r at a time, when k particular object are always to be
(i) included is n-kCr-k
(ii) excluded is n-kCr
(iii) inclded and s particular things are to be excluded is n-k-sCr-k
8. Total number of combinations in different cases-
(i) The number of selection of n identical object , taken at least one = n
(ii) The number of selections from n differdnt objects, taken at least one
= nC1 + nC2 + nC3 +
. + nCn = 2n - 1
(iii) The number of selections of r objects out of n identical object is 1.
(iv) Total number of selections of zero or more object form n identical object is n+1
(v) Total number of selection of zero or more objects out of n different objects
= nC0 + nC1 + nC2 + nC3 +
..+ nCn = 2n
(vi) The total number of selections of at least one of a1 + a2 +
.. + an object where a1 are alike )of one kind ), a2 are alike (of second kind) ,
an are
[(a1 + 1) (a2 + 1)(a3 + 1) +
. + (an + 1)] -1
(vii) The number of selections taking atleast one out of a1 + a2 + a3 +
+ an k object when a1 are alike (of one kind), a2 are alike (of second kind),
.. an are alike (of kth kind) and k are distinct is [(a1 + 1) (a2 +1) (a3 + 1)
. (an + 1) ] 2k - 1
9. Division and distribution-
(i) The number of ways in which (m+n+p) different objects cen be divided into there groups containing m, n, & p different objects respectively is
(ii) The total number of ways in which n different objects are to be divided into r groups of group size of no two groups
(iii) The total number of ways in which n different object are to be divided into groups such that k1 groups have groups size n1 , k2 groups have groups size n2 and so on kr groups have size nr , is given as
(iv) The total number of ways in which in n different objects are divided into k groups of fixed group size and are distributed k persons (one group to each) is given as (number of ways of group formation ) x k!
10. Selection of light ht objects and multinomial theorem-
(i) The cofficient of xn in the expansion of (1-x-r) is equal to n+r-1Cr-1
(ii) The number of solution of the equation x1 + x2 +
.+ xr = n, n Î N under the condition n1 £ x1 £ n1, n2 £ x2 £ n2 ,
.. nr £ xr £ nr
Where all xis are integers is given as Coefficient of xn is
11. Derangement Theorem
(i) If n thing are arranged in a row, then the number of ways in which they can be rearranged so that no one of them occupies the place assigned to it is
(ii) If n things are arranged at n places then the number of ways to rearrange exactly r things at right places is
12. Some Important result
(a) Number of total different straight lines formed by joining the n points on a plane of which m(nC2 - mC2 + 1
(b) Number of total triangles formed by joining the n points on a plane of which m(nC3 - mC3.
(c) Number of diagonals in a polygon of n sides is nC2 - n.
(d) If m parallel lines in a plane are intersected by a family of other n parallel lines. Then total number of parallelogram so formed is mC2 x nC2.
(e) Given n points on the circumference of a circle, then number of straight line nC2 number of tringles nC3 number of quadrilaterals nC4
(f) If n straight lines are drawn in the plane such that no two lines are parallel and no there lines are concurrent . then the number of part into which these lines divide the plane is = 1 + ån
(g) Number of rectangles of any size in a square of n x n is r=1ån r3 and number of squares of any size is r=1ånr2
(h) Number of rectangles of any size in a rectangle of n x p is (np/4) (n+1) (p+1) and number of squares of any size r=1ån (n+1-r)(p+1-r).
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