COUNTING RULES.

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Counting Principles –

There are two basic counting principles

1.Sum Rule

2.Product Rule.

▪Sum rule:-

Definition:- If an event can occur in n ways or m ways , where none of the set of n ways or m ways are dependent upon each other than the total number of ways in which the task can be done is n+m ways. 

Illustration 1

From city A to D there are two ways to go to city D via B or C. There are 9 ways to go via city B and 7 ways to go via city C.Find the total number of ways to go to city D from A.

Solution 

We can go by 9+7 ways

i.e 16 ways

▪Product Rule:-

Definition:-If a task can be broken down into a sequence of k subtask , such that each subtask can be performed in n1,n2,n3.......nk ways then the task can be performed in n1×n2×n3...×nk ways.

 

Illustration 1

In how many ways can 3 winning prizes be given to the top 3 players in a game played by 15 players?

Solution 

We have to distribute three prizes so there are three sub tasks.

1st prize might be given to any of 15 players. 

2nd prize will be given to 14 players. 

3rd prize will be given to 13 players.

So total number of ways are

15×14×13=2730 ways

Illustration 2

How many distinct 4-digit number greater than 1000 can be formed by rearranging the digits of 2020?

Answer =>3

Solution 

For the digit at thousandths place it should be greater than 0

We have only one distinct digit (2) greater then 0.

For next digits at remaining three places we have either two options (0 or 2)

There are 3 possibilities 002 or 020 or 200

Total cases

1×3=3

Illustration 3

How many positive integers with 5 digits can be made up using the digits 1, 2 and 3.

Solution (with explanation)

Suppose we look for all the 2 digit numbers first then work up to 5. We can make a list:

11, 12, 13, 21, 22, 23, 31, 32, 33. So there are 9.

The reason for this seems to be that there are three numbers that can go in the first place. For every

number in the first place there are three numbers that can go in the second place. 3×3 = 9.

Right then, let's tackle 5 digits. There are 3 choices for the first place, 3 for the second, 3 for the

third, 3 for the fourth and 3 for the fifth. Altogether we've got 3 × 3 ×3 × 3 × 3 = 3⁵=243

Written by:- Team Mathfreaks.

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