# Statistics: Counting Techniques

Problem: How many words can be formed by ordering the letters ARKANSAS ?

First we need to count the number of letters or spots in our word ARKANSAS. The word contains 8 letters.

Now we want to look at the number ways we can arrange one of the letters in the 8 spots, let’s start with the letter ‘A’ in ARKANSAS. We first need to count the number of A’s, the number of A’s = 3. Now we can use the combination formula where n=8 and k=3.

So the combination formula for the A’s is nCk or 8C3, where again n=8 and k=3. More importantly if we solve the formula then 8C3 = 56, which is the number of ways we can arrange the letter A.

The number of ways we can arrange the 3 A’s in 8 spots
8C3 = 56

Now, we only have 5 spots left, because we used 3 of them for the letter ‘A’, so let’s see how we can place any two of the ‘S’ character in the remaining 5 positions. Again we must use the combination formula, but this time n= 5, and k = 2 to get 5C2.

The number of ways we can arrange the 2 S’s in 5 spots
5C2 = 10

Next let’s see how we can arrange the one ‘R’ in the remaining 3 positions. Only 3 positions remain because we have used the 2 S’s and the 3 A’s, so 5 spots out of 8 have been taken.

The number of ways we can arrange the 1 R in 3 spots
3C1 = 3

Next we want to arrange the one ‘K’ in any one of the remaining two positions.

The number of ways we can arrange the 1 K in 2 spots
2C1 = 2

Last but not least let’s place the one ’N’ in any one of the remaining one positions.

The number of ways we can arrange the 1 N in 1 spot
1C1 = 1

## CONCLUSION:

The original question asked, how many words can be formed by ordering the letters ARKANSAS ?

The answer is (56) (10) (3) (2) (1) = 3,360.

So there are 3,360 words that can be formed by ordering the letters in ARKANSAS.

Below, you can watch another video on counting techniques !

If you want to read up on more combinatoric problems like the one in this article or Discrete Math topics in general a great book to easily learn and practice these topics is Practice Problems in Discrete Mathematics by Bojana Obrenic’, and Discrete Math Workbook: Interactive Exercises by James R. bush.

# Check Out the following for content / videos on Computer Science, Algorithm Analysis, Programming and Logic:

compsci112358:

Video Tutorials on Recurrence Relation:

Video Tutorial on Algorithm Analysis:
https://www.udemy.com/algorithm-analysis/

Written by