Problem 18 - Algebra and Trigonometry

Question

13–18 Evaluate the expression. $$C(99,3)$$

Step-by-Step Solution

Answer

The value of \(C(99, 3)\) is 156849.

Step 1: Understand the problem

We are asked to evaluate the expression \(C(99,3)\), which means we need to calculate the number of combinations of 99 items taken 3 at a time.

Step 2: Recall the formula

The formula for combinations \(C(n, r)\) is given by \[C(n, r) = \frac{n!}{r!(n-r)!}\]Here, \(n\) is 99 and \(r\) is 3.

Step 3: Substitute values into the formula

Substitute \(n = 99\) and \(r = 3\) into the combination formula:\[C(99, 3) = \frac{99!}{3!(99-3)!}\]

Step 4: Calculate factorials

Calculate the factorials needed:- Find \(3! = 3 \times 2 \times 1 = 6\)- Find \(99!\) and simplify with \(96!\) in the denominator: \(99! = 99 \times 98 \times 97 \times 96!\).- The \(96!\) will cancel out from the numerator and the denominator.

Step 5: Perform the division

Substitute the simplified factorial expression back into the combination formula:\[C(99, 3) = \frac{99 \times 98 \times 97}{3 \times 2 \times 1}\]- Simplify the division step-by-step: - Divide \(99 \times 98 \times 97\) by 6 - This simplifies to 156849.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics and are often used in permutations and combinations. A factorial, denoted by an exclamation mark (e.g., \(!n!\)), represents the product of all positive integers from 1 up to \(n\). For example, \(3! = 3 \times 2 \times 1 = 6\).
The factorial of a number \(n\) is the total number of ways to arrange \(n\) distinct objects into a sequence.
Here are some key points about factorials:

• Factorial of zero is always 1, i.e., \(0! = 1\).
• Factorials grow very quickly with larger numbers.
• Used mainly in permutations, combinations, and probability calculations.

Calculating factorials involves multiplying a series of descending natural numbers. This operation can quickly become large, which is why simplifications like cancellation are crucial.

Combination Formula

The combination formula is used to determine how many ways we can choose \(r\) objects from a set of \(n\) objects, where the order does not matter.
The mathematical representation of the combination formula is:\[C(n, r) = \frac{n!}{r!(n-r)!}\]Here, \(n!\) signifies the factorial of \(n\), \(r!\) is the factorial of \(r\), and \((n-r)!\) is the factorial of the difference between \(n\) and \(r\).

• No Repetition: An object can be chosen only once in each combination.
• Order Does Not Matter: The sequence in which items are chosen does not change the combination.

Let's see how it works with the example \(C(99, 3)\). This means choosing 3 items out of 99 available. After simplifying, we calculate using the following steps:

• Compute \(99!\)
• Break it into a manageable form \(99 \times 98 \times 97 \times 96!\)
• Cancel \(96!\) from the numerator and the denominator to simplify calculations

Probability

Probability, in simple terms, is the measure of the likelihood that an event will occur. It's expressed as a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty.
In the context of combinations, probability can be calculated by determining the favorable number of outcomes (using combinations) over the total number of possible outcomes.

• Express a probability as a fraction \(\frac{favorable \ outcomes}{total \ outcomes}\).
• Use combinations to find the total ways of selecting subsets, thus aiding in calculating probabilities in scenarios like card games or lotteries.

Connecting back to combinations, if you know how many ways you can select a group of items, you can predict the likelihood of selecting any specific group. This applies to various fields such as statistics, finance, game theory, etc.

Binomial Coefficients

Binomial coefficients are coefficients in the expanded form of a binomial expression. They are expressed using the combination formula \(C(n, r)\) as follows:\[(n + k)^n = \sum_{i=0}^{n} C(n, i)a^{n-i}b^i\]Here, each coefficient is a binomial coefficient.
The function \(C(n, i)\) signifies the number of combinations, which are used as coefficients in the expansion of powers of a binomial.

• Used in binomial theorem for providing the coefficient of the terms in the polynomial expansion.
• Hugely useful in areas like algebra and probability theory.
• The triangle of binomial coefficients is often depicted in Pascal's Triangle, showing the symmetrical properties of these coefficients.

In our exercise \(C(99, 3)\), it serves as a binomial coefficient for a term in expansion when raised to a power.