how many words consisting of 5 consonants and 3 vowels be formed out of 7 consonants and 5 vowels
There are 7C5 ways to pick the consonants
There are 5C3 ways to pick the vowels
Each set of 8 letters can then be rearranged in 8! ways.
This is assuming no duplicate letters.
To find the number of words consisting of 5 consonants and 3 vowels that can be formed out of 7 consonants and 5 vowels, we can use the concept of combinations.
First, let's calculate the number of ways to choose 5 consonants from the given 7 consonants. This can be done using the combination formula, nCr, where n represents the total number of items and r represents the number of items to be chosen.
The number of ways to choose 5 consonants from 7 can be calculated as:
C(7, 5) = 7! / [(5!)(7-5)!]
= 7! / [(5!)(2!)]
= (7 × 6 × 5!) / [(5!)(2 × 1)]
= (7 × 6) / (2 × 1)
= 21
Thus, there are 21 ways to choose 5 consonants from the given 7 consonants.
Next, let's calculate the number of ways to choose 3 vowels from the given 5 vowels.
C(5, 3) = 5! / [(3!)(5-3)!]
= 5! / [(3!)(2!)]
= (5 × 4 × 3!) / [(3!)(2 × 1)]
= (5 × 4) / (2 × 1)
= 10
Therefore, there are 10 ways to choose 3 vowels from the given 5 vowels.
Finally, to find the total number of words, we multiply the number of ways to choose the consonants by the number of ways to choose the vowels:
Total number of words = 21 × 10
= 210
Hence, there are 210 words consisting of 5 consonants and 3 vowels that can be formed out of 7 consonants and 5 vowels.