In this worksheet, let us learn how to solve a problem on permutations.
By learning to solve this question in permutations and combinations topic, you will learn how to distinguish permutation from combination using the criteria of order.
(recall that ORDER is the criteria that distinguishes permutations from combinations)
Here goes the question:
SOLVED PROBLEM 1:
There are ‘n’ persons In a meeting. A total number of 420 cards will be exchanged if each one exchanges a card with every other in the meeting. Find how many persons are there in the meeting.
As given, take the number of persons in the meeting as ‘n’.
Just imagine whether the action of exchanging a card by one person with every other underlies which of the two concepts:
Of course it is permutations.
Suppose a person called A exchanges a card with another person B. let us call this card as AB.
Now, let the person B also exchange a card with A.
let us now call this card exchanged as BA.
Let us ask the following question at this stage:
Are the two cards AB and BA exchanged by each of A, B with the other same or different?
They must be different!
But, how and why?
Just imagine A and B as two employees of an office. If A gives his own visiting card to B and B, his own to A, then the two cards exchanged between AB and BA are different or same?
Ofcourse, they are different, because each staff has his or her own visiting card to give to others.
And remember, AB and BA are deemed different when you treat them from permutation’s point of view because order holds significance in permutations and order is the underlying principle distinguishing AB from BA.
(to understand better and in more detail as to how AB and BA are deemed different in permutations, do check out the following link in our website www.math-for-all-grades.com:
Since the two cards AB and BA are different from each other, the action of exchanging of cards by each person in the gathering with every other must be concluded as an instance of permutation.
Now, after having determined the act of exchanging cards as Permutations, let us now proceed with the remaining part of the solution:
As given in the question, the number of cards exchanged by the ‘n’ persons is 420.
Well and fine, be that as it may.
But what will you write for the total number of cards exchanged among the ‘n’ persons in terms of ‘n’, which we are supposed to find in the question.
If you have not guessed it, if you have not found it out as yet, then the answer for the total number of cards exchanged among the ‘n’ persons, when each one exchanges a card with every other will be:
n × (n – 1)
How is it n × (n – 1)
now, let me give you a simpler instance of cards exchanged among 3 persons.
What will be the total number of cards exchanged among 3 persons, if each one exchanges a card with every other?
It would also be 3 × (3 – 1) = 3 × (2) = 6.
But how is it?
Now, among 3 persons say A, B and C, if each one gives a card to every other, then the exchange of cards among these 3 persons could be visualized in the following way:
The person A can give a card to the two others B and C.
thus by A 2 cards will be given away. You can call the 2 cards as AB and AC
Next, B can give a card to each of the other two A and C in which way thus B will give away 2 cards which can be named as BA and BC.
So, by the two persons A and B, a total of 4 cards have got exchanged.
(the two cards AB and BA are indeed different! Think for yourself)
C can give away one card to each of A and B and thus exchange another 2 cards:
CA and CB.
Thus among the three persons A, B and C a total of 6 cards will be exchanged.
This is what we wrote as:
3 × (3 – 1) = 3 × (2) = 6.
Generalize this instance of 6 cards exchanged among 3 persons expressed as
3 × (3 – 1)
Tell me, what can we write the expression for the total number of cards exchanged among ‘n’ persons when each one exchanges a card with every other. You’re right! It is indeed:
n × (n – 1).
n × (n – 1) = 420
now you have a quadratic equation to be solved to find ‘n’ i.e. the number of persons in the meeting.
If you are not familiar with skill of solving a quadratic equation, then do click the link below:
Let us now find ‘n’ from the equation:
n × (n – 1) = 420
n2 – n = 420
n2 – n – 420 = 0
now to solve for the variable ‘n’ , think of two numbers whose sum is the coefficient -1 before n in the above equation and whose product is the number -420 in the above quadratic equation.
The two numbers whose sum is -1 and product is -420 are 21 and 20.
You will have to take the greater number 21 as negative and smaller number 20 as positive so that their sum -21 +20 = -1 will be satisfied and the product -21 × 20 = -420 also will be satisified.
The following expression will therefore be the factorization step of the above quadratic equation: n2 – n – 420 = 0
n2 – 21n + 20n – 420 = 0
n(n – 21 ) +20( n – 21) = 0
i.e. (n – 21 ) (n + 20) = 0
now equate each factor to zero to find the roots of the variable ‘n’ as below:
n – 21 = 0 or n + 20 = 0
i.e. n = 21 or n = -20
since n denotes the number of persons which cannot be a negative number, therefore the value of -20 obtained for n will be discarded and the other one 21 for is indeed a sensible and acceptable value.
So, the number of persons in the meeting were 21.
You could have solved this question by applying the formula:
to learn about the formula above, click the following link:
in order to apply the formula nPr
you must determine that order of operation holds importance. But how?
Of course, the discussion whether order matters has been dealt in detail above, so I request you to go back to the above lines where the reason behind order having significance in this question has been discussed in detail.
Set the number of persons in the meeting as ‘n’.
You will need 2 persons for exchange of one card.
In this way the number of cards exchanged among the n persons will be
which is equal to 420.
i.e. nP2 = 420.
[n × (n – 1) × (n – 2)!]/(n – 2)! = 420
i.e. [n × (n – 1) × (n – 2)!]/(n – 2)! = 420
n × (n – 1) = 420
again, find out the value of n by solving the above quadratic equation as done in the above lines.