Permutation And Combination Problems
Permutation And Combination Problems
When the order doesn't matter, it is a Combination. When the order does matter it is a Permutation. In other words A Permutation is an ordered Combination.
- Combination When the order doesn't matter, it is a Combination. A Permutation is an ordered Combination.
- Permutation When the order does matter it is a Permutation. To help you to remember, think "Permutation ... Position"
The Different arrangements which can be made by taking some or all of the given things or objects at a time is called Permutation.
Each of the different selections or groups which is made by taking some or all of a number of things or objects at a time is called combination .
Types of Permutation
There are basically two types of permutation:
- Repetition is Allowed: such as the lock above. It could be "333".
- No Repetition: for example the first three people in a running race. You can't be first and second.
Permutations with Repetition
These are the easiest to calculate. When a thing has n different types ... we have n choices each time! For example: choosing 3 of those things, the permutations are: n * n * n (n multiplied 3 times)
More generally: choosing r of something that has n different types, the permutations are: n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilities for the second choice, and so on, multiplying each time.)Which is easier to write down using an exponent of r: n × n × ... (r times) = n r
Example: Iin the lock above, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them: 10 × 10 × ... (3 times) = 103 = 1,000 permutations
So, the formula is simply:
nr where n is the number of things to choose from, and we choose r of them, repetition is allowed, and order matters.
Question 1: In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
- (A) 920
- (B) 825
- (C) 720
- (D) 610
Answer: (C) 720
The word 'OPTICAL' has 7 letters. It has the vowels 'O','I','A' in it and these 3 vowels should always come together. Hence these three vowels can be grouped and considered as a single letter. That is, PTCL(OIA). Hence we can assume total letters as 5 and all these letters are different. Number of ways to arrange these letters = 5! = 5 * * 3 * 2 * 1 = 120 All the 3 vowels (OIA) are different Number of ways to arrange these vowels among themselves = 3! = 3* 2 * 1 = 6 Hence, required number of ways = 120 * 6 = 720
Question 2: In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
- (A) 810
- (B) 1440
- (C) 2880
- (D) 50400
Answer: (D) 50400
In the word 'CORPORATION', we treat the vowels OOAIO as one letter. Thus, we have CRPRTN (OOAIO). This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different. Number of ways arranging these letters = 7!/2! = 2520 Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in 5!/3! = 20 ways Required number of ways = (2520 x 20) = 50400.
Question 3: In how many ways can the letters of the word 'LEADER' be arranged?
- (A) 72
- (B) 144
- (C) 360
- (D) 720
Answer: (C) 360
The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R. Required number of ways = 6!/(1!)(2!)(1!)(1!)(1!) = 360
Question 1: In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?
- (A) 32
- (B) 48
- (C) 36
- (D) 60
Answer: (C) 36
There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants. Let us mark these positions as under: (1) (2) (3) (4) (5) (6) Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5 Number of ways of arranging the vowels = 3P3 = 3! = 6 Also, the 3 consonants can be arranged at the remaining 3 positions. Number of ways of these arrangements = 3P3 = 3! = 6 Total number of ways = (6 x 6) = 36.