In this article, we will discuss the basic concepts and formulas of Permutation & Combination required for solving problems in various placement entrance tests and competitive exams.The questions from this topic are mainly focused on checking the skill of an aspirant in logical counting.
Learn the fundamental principles of Permutation and Combination along with solved examples.
Formula's for Permutation and Combination
Let n be a positive integer. Then, factorial n, denoted n! is defined as:
n! = n(n - 1)(n - 2) ... 3.2.1.
- We define 0! = 1.
- 4! = (4 x 3 x 2 x 1) = 24.
- 5! = (5 x 4 x 3 x 2 x 1) = 120.
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
- All permutations (or arrangements) made with the letters a, b, c by taking two at a time are
(ab, ba, ac, ca, bc, cb).
- All permutations made with the letters a, b, c taking all at a time are:
( abc, acb, bac, bca, cab, cba)
Number of Permutations:
Number of all permutations of n things, taken r at a time, is given by:
nPr = n! / (n - r)!
- 6P2 = (6 x 5) = 30.
- 7P3 = (7 x 6 x 5) = 210.
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
- Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.
- Note: AB and BA represent the same selection.
- All the combinations formed by a, b, c taking ab, bc, ca.
- The only combination that can be formed of three letters a, b, c taken all at a time is abc.
- Various groups of 2 out of four persons A, B, C, D are:
- AB, AC, AD, BC, BD, CD.
- Note that ab ba is two different permutations but they represent the same combination.
Number of Combinations:
The number of all combinations of n things, taken r at a time is:
nCr = n! / (r !) (n - r) !
- nCn = 1 and nC0 = 1.
- nCr = nC(n-r)
11C4 = (11 x 10 x 9 x 8) / (4 x 3 x 2 x 1) = 330.
16C13 = 16C(16-13) = 16C13 = (16 x 15 x 14) / (3 x 2 x 1) = 560.
Practice Questions For Permutation and Combination
Q1)An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, …., 9 such that the first digit of the code is nonzero. The code, handwritten on a slip, can however potentially create confusion, when read upside down-for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?
Q2) If a refrigerator contains 12 cans such that 7 blue cans and 5 red cans. In how many ways can we remove 8 cans so that at least 1 blue can and 1 red can remains in the refrigerator.
Q3) There is meeting of 20 delegates that is to be held in a hotel. In how many ways these delegates can be seated along a round table, if three particular delegates always seat together?
A) 17! 3!
B) 18! 3!
C) 17! 4!
D) 18! 4!
Q4) In how many ways a four digit even number can be formed by using the digits 2,3,5,8 exactly once?
Q5) A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?
C) 10C6 * 6!
D) 10C5 * 6
Q6) Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
Q7) A boy has nine trousers and 12 shirts. In how many different ways can he select a trouser and a shirt?
Q8) Find the sum of all the 4 digit numbers that can be formed with the digits 3, 4, 5 and 6
D) None of these
Q9)In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?
Q10) How many combinations of students are possible if the group is to consist of exactly 3 freshmen?