24
How many 5-letter “words” can be formed from the letters of the word FORMULATED, if each must contain 2 vowels and 3 consonants? How many of these will have the vowels at the two ends on the “word”?
To calculate the number of 5-letter "words" that can be formed from the letters of the word FORMULATED, if each must contain 2 vowels and 3 consonants, we need to follow these steps:
Step 1: Determine the number of vowel combinations
The word FORMULATED contains 5 vowels (O, U, A, E, and I). We need to choose 2 vowels from these 5 to include in our 5-letter word. This can be done using a combination formula. The number of combinations of choosing 2 objects from a set of 5 is given by:
C(5,2) = 5! / (2! * (5-2)!) = 10
So, there are 10 different combinations of 2 vowels that can be chosen.
Step 2: Determine the number of consonant combinations
The word FORMULATED contains 6 consonants (F, R, M, L, T, and D). We need to choose 3 consonants from these 6 to include in our 5-letter word. Using the combination formula, the number of combinations of choosing 3 objects from a set of 6 is:
C(6,3) = 6! / (3! * (6-3)!) = 20
Therefore, there are 20 different combinations of 3 consonants that can be chosen.
Step 3: Multiply the number of vowel combinations and consonant combinations
To find the total number of 5-letter words, we need to multiply the number of vowel combinations (10) by the number of consonant combinations (20):
Total number of words = 10 * 20 = 200
So, there are 200 different 5-letter "words" that can be formed from the letters of the word FORMULATED if each must contain 2 vowels and 3 consonants.
Now, to determine how many of these words will have the vowels at the two ends of the "word", we can consider the positions of the vowels in a 5-letter word as follows:
First position: a vowel
Second position: any letter (can be a vowel or a consonant)
Third position: any letter (can be a vowel or a consonant)
Fourth position: any letter (can be a vowel or a consonant)
Fifth position: a vowel
In this case, we have 5 different vowel options for the first and fifth positions. For the second, third, and fourth positions, we can choose from any of the remaining consonants and vowels. Since we already know that we need to choose 2 vowels and 3 consonants, we can calculate this as:
Number of words with vowels at the two ends = 5 * (number of combinations of 3 consonants from 6) = 5 * C(6,3) = 5 * 20 = 100
Therefore, there are 100 different 5-letter "words" that can be formed from the letters of the word FORMULATED, which have the vowels at the two ends of the "word".
Is 24 your answer?
Which words did you find?