How many arrangements of the letters in SCIENCE both begin and end with a vowel?

Question

...using a method used in Algebra 2?

Answer 1

There are three possibilities:

1. The word starts with an E and ends with an E. There is only one way that can happen, so we should only be concerned with the other letters. The number of ways you can rearrange the 5 letters in between is 5!/2! = 60(You have to divide by 2! because there are two C's. Rearranging their positions would lead to the same arrangement.)

2. The word starts with an E and ends with an I. There is again 5!/2! = 60 ways to rearrange the other letters.

3. The word starts with an I and ends with an E. There are 60 ways to rearrange the other letters.

So in total, there are 60+60+60 = 180 arrangements of the letters in SCIENCE that both start and end with a vowel.

edit: Judy, 180 is the correct answer. There is no reason to say that the C's are different according to the problem. Your good.

Answer 2

The arrangement can either start with an I or an E, and can end with an I or an E, so there are three possibilities for the starting and ending letters - I.....E, E.....E, or E.....I. There are five letters in the middle - there are 5! or 5*4*3*2*1, or 120 possible permutations for those. You'd have to divide that number by 2 though, to account for the two C's, so there are really only 60 sequences for the middle five letters for each of the three start/end combinations. So there are 3 * 60 or 180 different arrangements of the letters that would begin and end with a vowel.

Warning: this might or might not be the answer the teacher is looking for, depending on whether they are considering the two C's as equivalent.