In how many ways 5 persons can be arranged in a line if a particular person

Calculating number of ways of dividing people or distinct objects into groups of the same size often pose difficulty to students.

Hopefully these examples will help students understand the concept.

Basic Example

How many ways can 10 children be divided into two groups of 5?

In how many ways 5 persons can be arranged in a line if a particular person

When dividing people or distinct objects into groups, if p groups are indistinguishable, we need to divide by p! to avoid double counting.

Challenging example 1 

At a particular reception, 9 guests are to stand at 3 identical round tables. How many ways can this be done if there are at least two people at each table?

In how many ways 5 persons can be arranged in a line if a particular person

Challenging example 2 (SAJC 2011/MYCT/2/6)

In how many ways 5 persons can be arranged in a line if a particular person

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Circular permutations often pose some difficulty to students. Let’s consider the following scenarios:

Scenario A: 10 people to be seated at a round table with 10 identical seats

Number of ways = (10-1)! = 9! = 362880

Scenario B: 5 people to be seated at a round table with 10 numbered seats

Number of ways = 10 P 5 = 30240

Scenario C: 6 people to be seated at a round table with 10 identical seats

In how many ways 5 persons can be arranged in a line if a particular person

There will be 4 identical empty seats. Consider fixing 1 occupied seat, and permutate the other 9 seats around it. There are 4 identical seats among the 9 seats. So number of ways = 9!/4! = 15120

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A tennis club has n male players and n female players. For a tournament the players are to be arranged in n pairs, with each pair consisting of one male and one female. Find the number of possible pairings.

Answer: n!

Solution

Suppose the male got to choose his partner. The first male can choose from n females. After he has chosen, the next male can choose from (n-1) females. So the number of possible pairings is n.(n-1).(n-2)…1 = n!

For instance there are 2 male players (M1, M2) and 2 female (F1, F2) players.

One way to arrange 2 pairs of players: M1 F1 and M2 F2

Second way to arrange 2 pairs of players: M1 F2 and M2 F1

Number of ways to arrange 2 pairs of players is 2!

Comments

Some students will give the wrong answer n x n, which is the number of ways to choose a pair.

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In how many ways can 9 balls of which 4 are red, 4 are white and 1 black be arranged in a line so that no red ball is next to the black?

The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

Example

How many different ways can the letters P, Q, R, S be arranged?

The answer is 4! = 24.

This is because there are four spaces to be filled: _, _, _, _

The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!

  • The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is:

n!        .
p! q! r! …

Example

In how many ways can the letters in the word: STATISTICS be arranged?

There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are:

10!=50 400
3! 2! 3!

Rings and Roundabouts

  • The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)!

When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!

Example

Ten people go to a party. How many different ways can they be seated?

Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440

Combinations

The number of ways of selecting r objects from n unlike objects is:

In how many ways 5 persons can be arranged in a line if a particular person

Example

There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls?

10C3 =10!=10 × 9 × 8= 120
             3! (10 – 3)!3 × 2 × 1

Permutations

A permutation is an ordered arrangement.

  • The number of ordered arrangements of r objects taken from n unlike objects is:

nPr =       n!       .
          (n – r)!

Example

In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.

10P3 =10!
            7!

= 720

There are therefore 720 different ways of picking the top three goals.

Probability

The above facts can be used to help solve problems in probability.

Example

In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?

The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 .

Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.

How many ways can 5 people be arranged in a line?

1 Expert Answer This question is asking for the number of permutations of 5 people. 120 is your answer.

How many ways can one arrange 5 students in a line?

Solution : Total number of ways of arranging 5 children in a row = 5! = 120.

How many ways can 5 people line up in a row for a photograph?

So, arrangements of 5 persons can be done in 5! =120 ways.

How many ways can you arrange 5 persons in a circle?

In how many different ways can five people be seated at a circular table? So the answer is 24.