Determine the number of different six letter arrangements that can be formed with the letters GUIDEPOST if the last two letters must be vowels?
To determine the number of different six-letter arrangements that can be formed with the letters GUIDEPOST, with the condition that the last two letters must be vowels, we can follow these steps:
Step 1: Count the total number of available letters.
In this case, we have 9 letters in total: G, U, I, D, E, P, O, S, T.
Step 2: Determine the number of vowels among these letters.
Among the 9 letters, the vowels are U, I, E, O. We have 4 vowels.
Step 3: Determine the number of consonants among these letters.
Consonants are all the letters that are not vowels. In this case, we have 5 consonants: G, D, P, S, T.
Step 4: Calculate the number of possibilities for placing the vowels.
Since the last two letters must be vowels, we have to determine the number of ways we can select and arrange 2 vowels from the 4 available vowels.
This can be calculated using the formula for combinations, denoted as "C(n, r)":
C(4, 2) = 4! / (2! * (4-2)!)
C(4, 2) = 4! / (2! * 2!)
C(4, 2) = 4 * 3 / 2 * 1
C(4, 2) = 6
So there are 6 ways to place the vowels in the last two positions.
Step 5: Calculate the number of possibilities for arranging the remaining letters.
After placing the vowels, we have 7 remaining letters (G, D, P, S, T) and 4 positions to fill (the first four positions).
This can be calculated using the formula for permutations, denoted as "P(n, r)":
P(7, 4) = 7! / (7 - 4)!
P(7, 4) = 7! / 3!
P(7, 4) = 7 * 6 * 5 * 4 / 3 * 2 * 1
P(7, 4) = 840
So there are 840 ways to arrange the remaining letters.
Step 6: Multiply the number of possibilities from Step 4 and Step 5 together.
To determine the total number of different six-letter arrangements, we multiply the number of possibilities for placing the vowels and arranging the remaining letters.
Total = 6 * 840
Total = 5,040
Therefore, there are 5,040 different six-letter arrangements that can be formed with the letters GUIDEPOST, where the last two letters must be vowels.
To determine the number of different six-letter arrangements that can be formed with the letters GUIDEPOST, where the last two letters must be vowels, we can follow these steps:
Step 1: Count the number of vowels in the word GUIDEPOST.
There are 4 vowels: U, I, E, and O.
Step 2: Count the number of consonants in the word GUIDEPOST.
There are 5 consonants: G, D, P, S, and T.
Step 3: Determine the number of choices for the last two positions (vowels).
Since there are 4 vowels remaining and the positions can be filled by any of these 4 vowels, we have 4 choices for the 5th letter and 3 choices for the 6th letter.
Step 4: Determine the number of choices for the remaining positions (consonants).
The first four positions can be filled with any of the 5 consonants remaining. For the first position, we have 5 choices. For the second position, we have 4 choices. For the third position, we have 3 choices. And for the fourth position, we have 2 choices.
Step 5: Multiply the number of choices from Steps 3 and 4.
The number of choices for the last two positions is 4 × 3 = 12.
The number of choices for the remaining positions is 5 × 4 × 3 × 2 = 120.
Step 6: Multiply the results from Steps 3 and 4.
The total number of different six-letter arrangements that can be formed with the letters GUIDEPOST, where the last two letters must be vowels, is 12 × 120 = 1440.
Therefore, there are 1440 different six-letter arrangements that can be formed with the letters GUIDEPOST if the last two letters must be vowels.