How many different arrangements can be made using all of the letters in the word iowa?

1. Check the tree diagram in the attached picture.

2. The first Letter can be one of the letters I, O , W or A. Assume your first letter is I, then you have 3 choices for the second letter, that is O, W or A.

3. Assume you pick O, then you have 2 options for the third letter; W or A. 

4. If you pick A, then the last letter can only be W. 

5. the path described is the "red line path" in the picture; that is the arrangement IOAW.

6. Another path is the "dashed line path" IAOW.

7. We count 6 "paths", where I is the first letter, each "path" standing for one arrangement. 

8. 4*6=24 is the total number of paths, that is arrangements of the letters IOWA.

9. Remark: The first choice is 4, combined (multiplied) by 3 choices for the second letter, combined (multiplied) by 2 choices for the third and finally multiplied by 1.

10. Thus there are 4*3*2*1 arrangements of 4 (different) letters.