How many ways are there to arrange the word: MATHEMATICS, if the last letter can not be "H", and the first letter must be a vowel?
First of all there are 11 letters. Line them up by vowels and consonants.
A A E I (vowels)
C H M M S T T (consonants)
You know you must begin with a vowel and must not end with H.
See what you can come up with? (There is always the dictionary!)
Sra
3 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2, i think
I GOT IT!!!!!!!
[10!/(2!2!) - (9!/2!2!)]+[(2)(10!/2!2!2!) - [(2)(9!/2!2!2!)]=1632960
To find the number of ways to arrange the word MATHEMATICS with the given conditions, we need to consider the possible positions for the last letter (H) and the first letter (a vowel).
First, let's count the number of ways to arrange the remaining 10 letters (excluding the last letter H). We have 10 letters: M, A, T, E, M, A, T, I, C, S.
The number of ways to arrange these 10 letters is calculated by 10 factorial (10!). The factorial denotes the product of all positive integers less than or equal to a given number. So, 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800.
Now, let's consider the positions for the last letter (H). Since the last letter cannot be H, we have 10 positions available for it. So, there are 10 possibilities for the position of the letter H.
Next, let's consider the possible positions for the first letter, which must be a vowel (A or I). In the remaining 9 letters, we have 2 vowels (A's) and 7 consonants. The number of ways to arrange these 9 letters is 9!, similar to what we calculated before.
To find the number of ways to arrange the word with the given conditions, we multiply the number of ways to arrange the remaining 10 letters (excluding H), the number of positions for the last letter (H), and the number of positions for the first letter (A or I).
So, the total number of ways to arrange the word MATHEMATICS with the given conditions is:
10! x 10 x 9!
Calculating this expression gives us the final answer.