Good afternoon, students, today we will start with next topic, Permutations and
Combinations. In permutations and combinations we will talk about counting. We will know
how to count fast. In this we will study how to count quickly.
For example, in this row 8 students are sitting. One way of counting is 1, 2, 3, 4, 5, 6, 7 and 8.
Other way of counting is 4 rows, 2 students in each row, 4 into 2 is 8. So what I used is I used
multiplication as tool. In the same way to count quickly we can use many different kinds of
tools by which counting will be very fast. And if we want to count in lakhs and crore which
will be difficult but counting by technique it will be easy. So, this is what we will learn in this
chapter.
Permutations and Combinations
Let us talk about first very important thing fundamental principle of multiplication,
fundamental principal of multiplication. If an event can occur in ‘m’ different ways, following
which another event can occur in ‘n’ different ways, then the total numbers of ways of
simultaneous occurrence of all these events in a definite order is m into n. If an event can
occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways.
Then the total number of ways of simultaneous occurrence of all these events in a definite
order is m into n. If one work is done in ‘m’ way, following which means after that another
work can be done in ‘n’ way. Then simultaneously both, in same order means after one is
finished then second should be done. In this way, how many ways are there – m into n.
Simultaneously, means first one should be done and then second one also should be done.
We have to do not only one but both of them should be done. Total number of ways will be
m into n.
Let us take an example, say if from you house to reach Lakshya and you come by auto. You
come to Lakshya by auto. If there are 2 ways to reach from your house to the auto, say 1 and
2, and from the auto to reach Lakshya there are 3 different ways, a, b, c. So, to reach
Lakshya from your house, how many ways will be there? 1a, 1b, 2a, 2c. 1a, 1b, 1c, 2a, 2b, 2c.
To reach Lakshya from house how many different ways will be there – 6. So to arrive at this
6, how fast can we do this? What is the work, to come from house to Lakshya. It is divided in
2 different ways, from house to auto, from auto to Lakshya. To do the first work, there is 2
ways and from auto to reach Lakshya, 3 ways. To do both the work simultaneously how
many ways, 2 into 3 is 6. So, to reach from house to Lakshya how many ways are there? 6.
Let us take one more example, say from house to reach classroom. You come by auto. From
house to reach the auto 2 ways, from auto to reach Lakshya – 3 ways and from Lakshya to
reach classroom there are 2 ways. So, how many ways will be there to reach classroom from
your house? 1ap, 1aq, 1bp, 1bq, 1cp, how, 1cp, if you follow the first way from home to
auto, then auto came by c way and after reaching Lakshya you took p way to reach
classroom and in the same way 1cq. Now how much this is? 6. Similarly, 2ap, 2aq, 2bp, 2bq,
2cp, 2cq, total is 12. Now, to count this it took some time. Suppose if I use the formula. Total
work is divided in 3 parts. To do 1st work 2 ways, after doing 1st to do second work 3 ways
and after 2nd work to do 3rd work 2 ways. Hence total number of ways is 2 into 3 into 2 that is
12. I hope you people got it, fundamental principle of multiplication. If to do first work there
is ‘m’ ways, for 2nd work ‘n’ and for 3rd work ‘p’ then to do all the 3 simultaneously, one after
then other is m into n into p.
Let us take few more examples. In how many ways 5 people can be seated in a 5 seater car?
5 people, 5 seater car. In how many ways 5 people can be seated in a 5 seater car? How
many place is there for a to sit, 5. After doing first work, after making place for a, so, how
many seats are there for b to sit, a must have sat in one seat, 1 place has been used from 5.
Now how many seats are left for b? 4. How many seats for b to sit – 4. Now two seats are
used. So after making place for a and b, how many seats are left c? 3. After making a, b, c,
sit, now only 2 seats are left. For d 2 seats and for e 1 seat. To do all the work
simultaneously, what we will do, multiple. So, to make 5 people sit in a 5 seater car, there
are 120 ways. For a there are 5 places, after making place for a, for b there are 4 places. That
means if I had to make 5 people to sit one by then first made a to sit, then b, then c, then d
and e. What you always do is you break whole of the work into small jobs. To do all the
works you write it down and then you multiple them. When will it be applied, when you do
all the work simultaneously, one by one.
Let us take one more example. How many different words can be formed using letters of
word ‘DELHI’? How many different words can be formed after arranging word ‘DELHI?’ It will
be 5 letter words, because there is 5 letters. We should think we have to fill five places. To
fill the first place 5 ways, after the first place is filled any one letter will be used. How many
letters are left – 4, so to fill the second place, 4 ways because letters can’t be repeated, to fill
second place 4. After filling first 2 places, in 3 place 3, then 2 and then 1. So, what will be the
total after filling all the places? 120. We have to arrange them to make a word. We can’t add
new letter and also can’t repeat the existing letters. So, to fill first place 5 ways, after 1 letter
is used, so 4 letters are left. To fill the second place, 4 ways, now 2 letters have been used.
So, for the third place 3 ways, then 2, then 1.
Let us solve few more questions, question first of your sheet. How many 5 lettered words
can be formed by rearranging letters of word ‘knife’ such that they start and end with
vowels? Where all is it constant, in the beginning and at end, what will come here, vowels?
So, from where it is constant, start question from there. From where it is constant, start
question from there? In the first place, vowel will only come. How many ways are there to
fill first place? 2, i or e, first place.
