Permutations and Combinations, Chapter Notes, Class 11, Maths - JEE PDF Download

Permutation & Combination

(A) Fundamental Principle of counting :

If an event can occur in `m' different ways, following which another event can occur in `n' different ways, then total number of ways of simultaneous occurrence of both events in definite order is = m × n (can be extended to any no. of events)

(B) What's Permutation & Combination ?

(1) Permutation :   * Arrangement of things taken some or all at a time.

                             * Order of occurrence of events is important.

(2) Combination :  * Collection or selection of things taken some or all at a time.

                             * Order of occurrence of events in not important

Note : All GOD made things in general are treated to be different and all man made things are to be spelled whether like or different

(C) Factorial :

(1) n ! = = product of 1^st `n' natural numbers ⇒ n ! = 1 × 2 × .......... × n

(2) (n - 1)! = ⇒

(3) factorial of negative numbers is not defined.

(D) Useful Theorems :

T-1 : Number of permutations of `n' distinct things taken r' at a time

T-2 : Numbers of combinations/selections of `n' distinct things taken `r' at a time

Note :- Derived Identities :

(1) ⁿc_r + ⁿc_{r - 1} = ^{n + 1}c_r

(2) ⁿc_r = ⁿc_{n - r}

(3) if ⁿc_x = ⁿc_y ⇒ x = y or x + y = 0

(4) ⁿp_r = r ! . ⁿc_r

(5) (2n)! = 2ⁿ.n! {1.3.5........(2n - 1)}

(E) Formation of Groups :

(1) Number of ways of dividing (m + n) different things in two groups having `m' and `n' things are : ;

(i) If m = n, then number of groups =

(ii) If `2n' things are to be equally distributed among 2 persons then, no. of ways = × 2!

(2) III_y by (m + n + p) different things can be divided into 3 unequal groups is

(i) If all groups are equal then number of ways =

(ii) If `3n' things are to be equally distributed among 3 persons then, number of ways=

Note :- This can be extended to any number of groups.

(F) Permutation of alike objects :

Number of permutation of `n' things taken all at a time out of which

(1) `p' are similar and of one kind

(2) `q' are similar and of second kind

(3) and rest `r' are all different =

Note :- Be careful if you encounter the following language used in problems

→ Number of other ways

→ Number of ways of rearranging

→ If as many more words as possible

(G) Circular permutation :

(1) Number of circular permutations of `n' different things taken `r' at a time = ⁿc_r (r - 1)!

(2) If clockwise & Anti-clockwise arrangements are considered as same then, ⁿc_r

(3) Number of circular permutations of `n' things out of which `p' are alike and rest are different =

(H) Total number of combinations :

(1) Number of ways of selecting at least one thing out of `n' different things is

= ⁿc₁ + ⁿc₂ +.............+ ⁿc_n = 2ⁿ - 1

(2) Number of ways of selecting at least one thing out of (p + q + r +..........) things in which p are alike of one kind, q of second kind & so on is = (p + 1) (q + 1) (r + 1) ........... -1

(I) Number of ways in which N can be resolved as a product of 2 divisors :

(1) N = p^a . q^b ...........p & q are prime

=

(2) Number of ways in which `N' can be resolved as a product of 2 divisors which are relatively prime = 2^n-1 , where n → number of primes involved in prime factorization of N.

(J) Maximizing ⁿc_r :

(K) Dearrangement :

Number of ways in which `n' letters can be placed in `n' directed envelopes so that no letter goes into its own envelope is = n!

(L) Distribution of alike objects :

(1) Number of ways of distributing `n' identical things to `p' persons where each person can receive one, none or more things is = ^{n + p - 1}C_{p - 1}

(2) Number of ways of distributing `n' identical things to `p' persons where each person should receive at least one object is = ^{n - 1}C_{p - 1}

(M) Grid Problem :

Number of ways of reaching B,

starting from point A are = .