Permutation n Combination

This is one of the most dreaded topics that a candidate can face in the GMAT exam. But, there are limited patterns that are asked in GMAT and if we can master them then i don't think one will have any trouble in solving these type of questions on GMAT.

I just got hold of my notes and decided to share em with fellow bloggers {there are some good questions that i will paste at the end of this post}:

1 Permutation : Permutation means arrangement of things. The word arrangement is used, if the order of things is considered.

2. Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.

3. Example: Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4,

To form this number the digits have to be arranged. Different numbers will get formed depending upon the order in which we arrange the digits. This is an example of Permutation.

4. Now suppose that we have to make a team of 11 players out of 20 players, This is an example of combination, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed.

5. nPr = n!/(n-r)!

6. Note : Factorial of negative-number is not defined. The expression  –3! has no meaning.
7. Number of permutations of n-thing, taken all at a time, in which ‘P’ are of one type, ‘g’ of them are of second-type, ‘r’ of them re of third-type, and rest are all different is given by :-
(n!)/(p!*q!*r!)   

8. Circular permutations

(a) If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by  (n-1)!

(b) If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (n-1)!/2!

Restricted – Permutations

9(a) Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is to be always included in each arrangement:

r*( n-1) P(r-1)

9(b) Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is never taken:
(n-1) P(r)


10. nCr = nC(n-r) {general formula}

11. Total number of ways of selecting one or more things out of n different things

nC1 + nC2 + nC3 + ------------- + nCn = 2^n - 1

12. Total number of ways of selecting ZERO or More things out of n different things

(nC0 + nC1 + -----------------nCn) = 2^n


13. (nC0 + nC1 + -----------------nCn) = 2^n {general formula}

14. Number of ways of selecting zero or more things from ‘n’ identical things is given by :- n+1

15. Number of ways of selecting one or more things from ‘p’ identical things of one type ‘q’ identical things of another type, ‘r’ identical things of the third type and ‘n’ different things is given by :

 (p+1) (q+1) (r+1)(2^n – 1)

Some good questions to apply the theory

Q How many ways can 10 letters be posted in 5 post boxes, if each of the post boxes can take more than 10 letters?

Q How many numbers are there between 100 and 1000 such that atleast one of their digits is 6?

Q How many ways can 4 prizes be given away to 3 boys, if each boy is eligible for all the prizes?

Q How many alphabets need to be there in a language if one were to make 1 million distinct 3 digit initials using the alphabets of the language?

Q How many number of times will the digit '7' be written when listing the integers from 1 to 1000?

Q Badri has 9 pairs of dark Blue socks and 9 pairs of Black socks. He keeps them all in a same bag. If he picks out three socks at random what is the probability he will get a matching pair?

Q How many words of 4 consonants and 3 vowels can be made from 12 consonants and 4 vowels, if all the letters are different?

Q If the letters of the word CHASM are rearranged to form 5 letter words such that none of the word repeat and the results arranged in ascending order as in a dictionary what is the rank of the word CHASM?

Q When four fair dice are rolled simultaneously, in how many outcomes will at least one of the dice show 3?

Q There are 5 Rock songs, 6 Carnatic songs and 3 Indi pop songs. How many different albums can be formed using the above repertoire if the albums should contain at least 1 Rock song and 1 Carnatic song?