Question

In how many ways can 4 consonants be chosen from the letters of the word SOMETHING? (1) \({ }^{9} \mathrm{C}\), (2) \({ }^{6} \mathrm{C}\) (3) \({ }^{4} \mathrm{C}\) (4) \({ }^{4} \mathrm{C}\)

Short answer

Answer: There are 15 ways to choose 4 consonants from the word "SOMETHING."

Step-by-step solution

Identify the consonants

In the word "SOMETHING," the consonants are S, M, T, H, N, G. There are a total of 6 consonants in the word.

Use combinations formula

We have 6 consonants, and we need to choose 4 of them. The number of ways to do this can be calculated using combinations formula which is: C(n,r) = n! / (r!(n-r)!) where n is the number of elements in the set (in our case, 6 consonants) and r is the number of elements to choose (in our case, 4 consonants).

Calculate the combinations

Using the formula, we need to calculate the number of ways to choose 4 consonants from 6: C(6,4) = 6! / (4!(6-4)!) C(6,4) = 6! / (4!2!) C(6,4) = (6 × 5 × 4 × 3 × 2 × 1) / ((4 × 3 × 2 × 1) × (2 × 1)) C(6,4) = (6 × 5 × 4 × 3 × 2 × 1) / (24 × 2) C(6,4) = 720 / 48 C(6,4) = 15 Thus, there are 15 ways to choose 4 consonants from the word "SOMETHING." The correct answer is (2) \({ }^{6} \mathrm{C}\).

Key concepts

Consonants

Consonants are the letters in the alphabet that are not vowels. In the English language, vowels include 'A', 'E', 'I', 'O', and 'U'. All other letters are classified as consonants. In word puzzles, crossword puzzles, and various linguistic exercises, identifying consonants is a vital step.
In our example with the word "SOMETHING," we need to pick consonants out of the letters given. The word "SOMETHING" has nine letters, out of which six letters are consonants: 'S', 'M', 'T', 'H', 'N', and 'G'. If you're dealing with problems that involve counting or selecting consonants, always begin by identifying them from the word or set of letters provided.

Combination formula

The combination formula is essential in combinatorics, the branch of mathematics dealing with counting and arranging items. When you're tasked with choosing a specific number of elements from a set without concern for the order, this formula is what you use.
The formula for combinations is given as:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Where:

• \(n\) is the total number of items (elements) in the set.
• \(r\) is the number of items to choose.
• The symbol \(!\) denotes factorial, which is the product of all positive integers up to that number.

To solve the exercise, we applied this formula with \(n = 6\) (the number of consonants) and \(r = 4\) (the number of consonants we need to choose). Applying the formula, we determined there are 15 different ways to choose 4 consonants from 6.

Permutations and combinations

Understanding the difference between permutations and combinations is key when dealing with these kinds of mathematical problems. The primary distinction lies in the importance of order:

• Permutations: Concerned with the arrangement of elements, where the order does matter. For instance, arranging the letters 'ABC' can result in 'ABC', 'ACB', 'BAC', and more.
• Combinations: Focus on the selection of elements, where the order does not matter. In combinations, 'ABC' is treated the same as 'ACB', such as when choosing team members or in this case, selecting consonants.

In the word problem, we used combinations because we were only interested in the sets of consonants themselves, not the order in which they were chosen. Remembering this distinction between permutations and combinations helps solve various problems efficiently.