Find the number of diffrent 8 letter wprds firmed from the letters of the word triangle if rach word is to have vowels occupying the second , third , and fourth places?
To find the number of unique 8-letter words formed from the letters of the word "triangle" with vowels occupying the second, third, and fourth places, we can proceed as follows:
1. There are 3 vowels in the word "triangle" (i, a, e). We need to choose one vowel to be placed in the second position. We have 3 choices for this selection.
2. After choosing a vowel for the second position, we will have 2 remaining vowels for the third and fourth positions. We need to choose one of these 2 vowels for the third position. We still have 2 choices for this selection.
3. After selecting a vowel for the third position, we will have 1 remaining vowel for the fourth position. We have 1 choice for this selection.
4. After placing the vowels in the second, third, and fourth positions, we have 5 remaining consonants (t, r, n, g, l) that can be used for the remaining 5 positions. For each of the remaining 5 positions, we have 5 choices.
To find the total number of 8-letter words using the counting principle, we multiply the number of choices at each step:
Total number of 8-letter words = (Number of choices for the second position) * (Number of choices for the third position) * (Number of choices for the fourth position) * (Number of choices for each of the remaining 5 positions)
= 3 * 2 * 1 * 5^5
= 3 * 2 * 1 * 3125
= 18750.
Therefore, there are 18,750 different 8-letter words formed from the letters of the word "triangle" if each word has vowels occupying the second, third, and fourth places.
To find the number of different 8-letter words formed from the letters of the word "triangle" if each word is to have vowels occupying the second, third, and fourth places, we can follow these steps:
Step 1: Identify the vowels in the word "triangle". In this case, there are two vowels, "i" and "a".
Step 2: Select the vowels for the second, third, and fourth places. Since we have two vowels, we can choose them in 2! (2 factorial) ways, which is equal to 2.
Step 3: Identify the consonants in the word "triangle". In this case, there are five consonants, "t", "r", "n", "g", and "l".
Step 4: Arrange the remaining consonants and the chosen vowels. We have 5 consonants left, which can be arranged in 5! (5 factorial) ways. Additionally, we have already chosen the vowels for the second, third, and fourth places, so we don't need to consider their positions anymore.
Step 5: Multiply the number of ways from Step 2 and the number of ways from Step 4 to get the total number of different 8-letter words.
Therefore, the number of different 8-letter words formed from the letters of the word "triangle" with vowels occupying the second, third, and fourth places is:
2! × 5! = 2 × 120 = 240.
So, there are 240 different 8-letter words that can be formed.
To find the number of different 8-letter words formed from the letters of the word "triangle" if each word is to have vowels occupying the second, third, and fourth places, we can follow these steps:
1. Identify the vowels in the word "triangle": The vowels in "triangle" are "i", "a", and "e".
2. Determine the number of choices for each position:
- Since the vowels must be in the second, third, and fourth places, there are 3 choices for the second position.
- For the third position, there are 2 remaining vowels to choose from.
- For the fourth position, after choosing the second and third positions, there is only 1 remaining vowel.
3. Determine the number of choices for the remaining positions:
- As the remaining 5 positions can be filled with consonants, we need to determine the number of consonants in "triangle".
- The consonants in "triangle" are "t", "r", "n", "g", and "l".
- Therefore, there are 5 choices for the first position, 4 choices for the fifth position, and so on until only 1 choice is left for the eighth position.
4. Calculate the total number of possible combinations:
- Multiply the number of choices for each position together to get the total number of combinations.
- 3 choices for the second position x 2 choices for the third position x 1 choice for the fourth position x 5 choices for the first position x 4 choices for the fifth position x 3 choices for the sixth position x 2 choices for the seventh position x 1 choice for the eighth position.
- The total number of combinations is: 3 x 2 x 1 x 5 x 4 x 3 x 2 x 1 = 720.
Therefore, there are 720 different 8-letter words formed from the letters of the word "triangle" with the condition that the vowels must occupy the second, third, and fourth places.