Question: Out of 10 consonants and 4 vowels, how many different words can be formed each containing 6 consonants and 3 vowels?

Solution:

Step 1: Determine the number of ways to choose 6 consonants out of 10.
To find the number of ways to choose 6 consonants out of 10, we can use the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of items and r is the number of items to be chosen.

Step 2: Determine the number of ways to choose 3 vowels out of 4.
Using the combination formula, we can calculate the number of ways to choose 3 vowels out of 4:
C(4, 3) = 4! / (3!(4-3)!)

Step 3: Determine the number of ways to arrange the chosen consonants and vowels.
Once we have selected the consonants and vowels, we need to arrange them in a specific order to form words. Since the order matters, we can use the permutation formula:
P(n, r) = n! / (n-r)!
where n is the total number of items and r is the number of items to be arranged.

Step 4: Calculate the total number of words.
To calculate the total number of words, we multiply the number of ways to choose the consonants, the number of ways to choose the vowels, and the number of ways to arrange the chosen consonants and vowels:
Total number of words = C(10, 6) * C(4, 3) * P(9, 9)

Step 5: Substitute the values into the formula and calculate the result.
Using the combination and permutation formulas, we can substitute the values into the formula and calculate the result:
Total number of words = (10! / (6!(10-6)!)) * (4! / (3!(4-3)!)) * (9! / (9-9)!)

Simplifying this expression, we get:
Total number of words = (10! / (6!4!)) * (4! / (3!1!)) * (9! / 1!)

Step 6: Calculate the result.
Using the factorial notation (n!) = n * (n-1) * ... * 1, we can calculate the result:
Total number of words = (10 * 9 * 8 * 7 * 6 * 5 * 4! / (6!4!)) * (4 * 3 * 2 * 1 / (3!1!)) * (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 / 1)
Total number of words = (10 * 9 * 8 * 7 * 6 * 5) * (4 * 3) * (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Total number of words = 151,200 * 12 * 362,880
Total number of words = 6,224,448,000