Understanding Consonant and Vowel Selection

The problem asks us to find the number of ways to form words using a specific number of consonants and vowels from a given set. We are provided with:

• A total of 5 consonants.
• A total of 4 vowels.

The task is to determine how many words can be made using exactly 3 consonants and 3 vowels. This process involves selecting the necessary letters from the available pool.

Calculating Letter Combinations

To find the total number of ways to select the required letters for the word, we need to calculate:

1. The number of ways to choose 3 consonants from the 5 available.
2. The number of ways to choose 3 vowels from the 4 available.

We will use the combination formula, denoted as $C(n, k)$ or $\binom{n}{k}$, which calculates the number of ways to choose $k$ items from a set of $n$ items without regard to the order of selection. The formula is:

$ C(n, k) = \frac{n!}{k!(n-k)!} $

Step 1: Selecting Consonants

We need to select 3 consonants out of 5. Using the combination formula:

$ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} $

Calculating the factorials:

$ C(5, 3) = \frac{5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times (2 \times 1)} = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 $

There are 10 ways to choose the 3 consonants.

Step 2: Selecting Vowels

Next, we need to select 3 vowels out of the 4 available.

$ C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} $

Calculating the factorials:

$ C(4, 3) = \frac{4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times 1} = \frac{4}{1} = 4 $

There are 4 ways to choose the 3 vowels.

Step 3: Total Number of Combinations

To find the total number of ways to select the group of letters (3 consonants and 3 vowels), we multiply the results from Step 1 and Step 2, according to the multiplication principle of counting.

Total number of ways = (Number of ways to choose consonants) $\times$ (Number of ways to choose vowels)

$ \text{Total Ways} = C(5, 3) \times C(4, 3) = 10 \times 4 = 40 $

Interpreting the Result

The calculation shows there are 40 different ways to select a group consisting of 3 consonants and 3 vowels. The wording "how many words... can be made" often implies arranging the selected letters. If arrangement were considered, we would multiply the 40 combinations by the number of ways to arrange the 6 selected letters ($6! = 720$), giving $40 \times 720 = 28800$ possible words. However, since 40 is one of the options and 28800 is not, it's understood that the question is specifically asking for the number of ways to select the group of letters, not the permutations of those letters into words.

Therefore, the total number of ways to make the selection is 40.