Question

Evaluate the combinations. $$ C_{1}^{20} $$

Short answer

Answer: The value of the combination \(C_{1}^{20}\) is 20.

Step-by-step solution

Identify n and k values

In our given question, we have \(C_{1}^{20}\). Here, n = 20 and k = 1.

Use the combination formula

Using the combination formula $$C_{k}^{n} = \dfrac{n!}{(k!(n-k)!)}$$, we need to plug in the values of n and k into the formula. So, $$C_{1}^{20} = \dfrac{20!}{(1!(20-1)!)}$$

Calculate factorials

Now, we'll calculate the factorials. 20! = 20 x 19 x 18 x ... x 2 x 1, 1! = 1, and (20 - 1)! = 19! = 19 x 18 x ... x 2 x 1.

Simplify the expression

Replace factorials in the formula: $$C_{1}^{20} = \dfrac{20!}{(1!(19)!)} = \dfrac{20 \times 19 \times 18 \times ... \times 2 \times 1}{(1 \times (19 \times 18 \times ... \times 2 \times 1))}$$ We can cancel out the common terms from the numerator and denominator: $$C_{1}^{20} = \dfrac{20}{1}$$

Final answer

After simplifying, we get the final answer: $$C_{1}^{20} = 20$$

Key concepts

Combination Formula

When it comes to selecting a subset from a larger set, combinations allow us to determine how many different groups can be formed. The combination formula is represented as \( C_{k}^{n} = \frac{n!}{k!(n-k)!} \) and it is used for calculating the number of combinations when selecting \( k \) items from \( n \) items without regard to order.

It's important to understand that in combinations, the order of selection doesn't matter. For example, selecting students for a committee from a class does not take into account the order in which they're selected.

• \( n \) is the total number of items to choose from.
• \( k \) is the number of items to be chosen.

In the exercise \( C_{1}^{20} \), it means we are selecting 1 item from a group of 20. All we need to do is plug the values into the formula to find the number of ways to make this selection. Often, this results in straightforward calculations.

Factorials

Factorials are a fundamental part of many combinatorial calculations, including the combination and permutation formulas. A factorial, denoted by an exclamation mark \( ! \), represents the product of all positive integers up to a given number \( n \). For example, \( n! \) is calculated as \( n \times (n-1) \times (n-2) \times ... \times 2 \times 1 \).

Factorials grow very quickly with larger numbers, which can make calculations complex. However, in many combinatorial problems, you'll notice that parts of the factorial in the numerator and the denominator cancel each other out, simplifying the math.

• \( 0! \) is defined as 1 by convention, because the product of no numbers at all is 1.
• Factorials are only defined for non-negative integers.

In the step-by-step solution above, calculating the factorials for \( 20! \), \( 1! \), and \( 19! \) is necessary to apply the combination formula correctly. Recognizing how these factorials fit into the formula helps streamline the process and avoid unnecessary calculation.

Permutations

Permutations and combinations are closely related concepts in combinatorics but have key differences. While both involve selecting items from a set, permutations are concerned with the order of selection, unlike combinations.

For permutations, order matters. The permutation formula is given by \( P_{k}^{n} = \frac{n!}{(n-k)!} \), where we are interested in arranging \( k \) items from \( n \) available items. This is different from combinations, where the formula involves dividing out \( k! \) since order doesn't matter there.

• Use permutations when the arrangement or sequence of the selected items is important.
• Commonly applicable in scenarios like seating arrangements, password generation, or races where sequence matters.

Although permutations were not explicitly needed in the provided exercise, understanding this concept is essential as it broadens your problem-solving toolkit in combinatorics. Learning to recognize when to apply permutations instead of combinations is a crucial skill in many mathematical problems.