© Pasang Dorji (Gesarling Central School, Dagana)
Fundamental Principal of Counting
Scenario 1:
Suppose you have 3 black shirts. Let us assume that you can wear only one shirt at a time. Then the number of ways
you can wear a t-shirt is 3, since you have three choices of t-shirts: 1st black shirt OR 2nd black shirt OR 3rd black
shirt. Pay attention to the word OR in this scenario:
1st OR (+)
2nd OR (+) 3rd
= 3 shirt choices
Additionally, if we have 4 white shirts, then you can wear a shirt in 7 ways in total as shown below:
1st OR 2nd OR
3rd OR
1st OR 2nd OR 3rd OR 4th
In short, you can wear any of the 3 white shirts OR any of the 4 black shirts which make 7 choices in total i.e.
3  4  7 ways.
Addition Rule: In general, if there are m ways of doing something, and n ways of doing
another thing, but cannot do after one thing, then there are m  n ways to perform
both of these operations.
Scenario 2:
Suppose you have 3 white shirts and 2 half pants. To dress up, you have to wear one shirt AND one half pant. The
number of ways you can dress up can be diagrammatically represented as follows:
1st
AND
1st
2nd
AND
OR
2nd
OR
AND
1st
3rd
AND
OR
2nd
OR
AND
1st
AND
OR
2nd
The above diagram can be interpreted as follows:



With the 1st shirt, you can either wear the 1st half-pant or the 2nd half-pant OR
With the 2nd shirt, you can either wear the 1st half-pant or the 2nd half-pant OR
With the 3rd shirt, you can either wear the 1st half-pant or the 2nd half-pant.
1
© Pasang Dorji (Gesarling Central School, Dagana)
We can see that for each choice of a shirt, there are two choices of pants, and thus, the total number of ways to dress
up is 6.
In short, we can dress up by wearing a shirt AND a half-pant. Since there are 3 shirts and 2 half-pants, the total
number of ways to dress up
Product Rule: In general, if there are m ways of doing something, and n ways of doing
another thing after one that, then there are m  n ways to perform both of these
operations.
Examples
1.
Dorji has 3 blue jackets, 4 black jackets, and 5 jeans. In how many ways can he dress up?
Solution: Since he can wear only one jacket or the other, there are 3  4  7 ways to wear a jacket. After wear
a jacket, he can wear any of the 5 jeans, so in total, there are 7  5  35 ways to dress up.
2.
There are 15 buses running between Thimphu and Phuntsholing. In how many ways can a man go from
Thimphu to Phuntsholing and return by a different bus?
Solution: To go from Thimphu to Phuntsholing, the man has 15 choices of bus. Since he cannot come by the
same bus he had taken, now, there are 14 choices of bus available for him to return back. So, in total there are
15 14  210 ways.
3.
How many four-letter words can be formed from 26 alphabets if a letter cannot be repeated?
Solution: In four-letter words, there are four slots to fill – there are 26 choices in the first slot, 25 in the second,
24 in the third and 23 in the fourth slot. So, in total 26  25  24  23  358, 800 words can be formed.
Try Your Self A4-I
1.
There are five sections in a school in which three new students have to be placed. In how many different ways
can we place them in the sections?
2.
Two friends hired a taxi in which there are 4 seats. In how many different ways can they seat themselves?
1.
ANSWERS
60
2. 12
2