Permutation And Combination Problems

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Ratio and Proportion Pipes and Cisterns

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"
permutation and combination problems

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

Solution

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

Solution

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

Solution

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

Solution

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.

Ratio and Proportion Pipes and Cisterns

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