If the following questions make you uncomfortable, then read this article.
- In how many ways can 5 member committee be formed out of 20 students? Or
- How many ways can the word “UPSC” be arranged so that “U” is always second from left.
- In a party of 20 guest, if every member shakes hands with other, how many total handshakes are done?
- Fundamental counting principle
- Permutation (Place, arrangement)
- Combination (choice)
- The Formulas
First the basics
- There are 3 trains from Mumbai to Ahmedabad and 7 trains from Ahmedabad to Kutch. In how many ways can Jethalal reach Kutch?
- From Mumbai to A’bad, Jethalal can go in 3 ways. (Because there are three train, he can pick anyone)
- Similarly From A’bad to Kutch, he can go in 7 ways.
- Total ways: simple multiplication =3 ways x 7 ways =21 ways Jethalal can reach Kutch.
- This is fundamental counting principle.
Let’s extend this further to sitting arrangement problems:
6 Men of Gokuldham society go to Abdul’s sodashop. And there are 6 chairs. Yes I’m talking about our beloved Jethalal, Bhide Master, Sodhi, Dr.Haathi, Mehta saab and Aiyyar.
In how many ways can they be seated? Or How many way can you arrange these 6 gentlemen into 6 chairs?
Let’s do one chair at a time.
For the first chair, you’ve 6 candidates:
Just pick one man and make him sit.
How many ways can you do it? Ans. 6 ways, bcoz you’ve 6 men and pick one. (Just like the train)
Now second chair: you’ve to pick one guy from the remaining 5 members. So 5 ways.
Third chair: 4 men remaining and you’ve to pick one: again 4 ways..
For the 6th chair only one man remains. So you can pick in 1 way only.
Now Just extending the fundamental counting principle
So total number of ways in which men of Gokuldham society can sit in chairs
= 6 x 5 x 4 x 3 x 2 x 1
=6! (six factorial ways)
How many arrangements possible?
= 6 x 5 x 4 x ..OK STOP! Because there are only 3 chairs.
First chair= 6 men pick one=6 ways.
Second chair = 5 men pick one=5 ways
Third chair=4 men pick one= 4 ways
So number of ways
In the 6 men 6 chairs problem, Dr.Haathi insists that he’ll sit in the number #1 chair only. Then How many arrangements are possible?
First chair= 1 man (Haathi) and you’ve to pick one= 1 way only!
Second chair=5 men remain, pick one =5 ways
third chair=4 men remain, pick one = 4 ways and so on…
So here we’ve
This is permutation. Here ‘order / ranking ’ matters. i.e. who sits in the first chair, who sits in the second chair etc.
The same question can appear under different wordings example
1. How many 4 lettered words can be formed out of “UPSC”, without repetition?
Answer: Same logic. Consider U, P, S, C are four gentlemen and they’ve to be arranged in 4 seats.
2. How many 4 letter words can be formed out of “UPSC” so that “U” always occupies the second position from left?
Answer: Same Dr.Haathi logic.
Second Topic is
The one and only Secretary of Gokuldham society, Master Bhide decides to form a 3 member-Committee for arrangement of Holi-festival. Out of the 5 members (Jetha, Sodhi, Mehta, Popat and Aiyyar) how many ways can he do this?
Here order or ranking doesn’t matter.
Because in case of chair sitting: we can say yes Mr.Sodhi is in first chair, Mehta saab in Second chair and so on…
But in case of Committee: it is only “IN or OUT” i.e. Yes Sodhi is in the Committee, No Mehta saab is not in the Committee.
So order doesn’t matter. Only selection matters.
How many ways can you pick up the first member? = 5 ways.
Second member? = 4 men remain, 4 ways
Third member? = 3 ways.
So total ways= 5 x 4 x 3= 60.
But wait, there is over counting.
A committee made up of Mehta, Sodhi and Aiyyar (MSA) is same as a Committee made up of Aiyyer, Mehta and Sodhi. (AMS) (because order doesn’t matter, only selection matters: are you “In or Out?”).
But in this answer 60, we’ve over counted such ‘orders’.
That’s why we need to divide the answer
Suppose Mehta, Sodhi and Aiyyar are “in” the committee.
Suppose they’ve to sit in three chairs. How many ways can you do it?
Just like permutation in first example
3 x 2 x 1=6.
That’s the overcounting : 6.
Our answer is
Handshakes is another type of combination problem because
Aiyyar shakes hand with Sodhi or Sodhi shakes hand with Aiyyer= both incidents are one and same. “order” doesn’t matter!
So How many ways can 6 men of Gokuldham society shake hands with each other?
To shake hands, you need two men.
How many ways can you pick up first man? = 6 ways.
How many ways can you pick second man? = 5 men remaining so 5 ways.
Hence answer is 6 x 5=30
But wait, over counting!!
Suppose Sodhi and Aiyyer (SA) are selected for handshaking: =two men.
How many ways can you make two men sit in two chairs?
= 2 x 1 =2 ways. (SA and AS)
That’s the over counting.
So we’ve to divide this over counting: to get the correct answer.
The books give readymade formulas:
But essentially they’re derived from above concepts of fundamental counting principle.
Permutation =place them = order matters in placement
Combination= choice = only “Yes or No” or “In or Out”
- Open your books right now, solve the sums using both the method shown above as well as the readymade formulas, then you’ll get a good command over these topics.
- Remember, You can never learn the aptitude by reading the ‘sums’.
- You must to get your hands dirty, and learn from the mistakes in calculation, so start solving the sums on paper by yourself.
- To be continued……
Well I hope I was able to explain the topic-concept to you.
But it is not sufficient to crack the exam! Still You might end up making silly mistakes in calculation or get stuck in some question during the actual exam. So You must practice as many sums as you can at home, to get a firm command over the topic. That’s why Get the books, and start solving sums!