Problem 61 - Elementary Algebra

Question

A box contains 10 computer chips. Three chips are to be chosen at random. The number of ways this can be done is $$\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$$

Step-by-Step Solution

Answer

Answer: 120 ways

Step 1: Identify n and r

In this problem, n = 10 since we have 10 computer chips. We are selecting 3 chips, so r = 3.

Step 2: Calculate factorials

Now, we need to calculate the factorials for n, r, and (n-r). The factorials are found by multiplying all positive integers up to the given number: 10! = 10×9×8×7×6×5×4×3×2×1 3! = 3×2×1 7! = 7×6×5×4×3×2×1 (since 10-3 = 7)

Step 3: Apply the combination formula

Now, we can apply the combination formula: C(10, 3) = 10! / (3!(7!))

Step 4: Plug the factorials into the formula

We will now substitute the calculated factorials in the combination formula: C(10, 3) = (10×9×8×7×6×5×4×3×2×1) / ((3×2×1)(7×6×5×4×3×2×1))

Step 5: Simplify the expression

To simplify the expression, we can cancel out the common factors in the numerator and the denominator: C(10, 3) = (10×9×8) / (3×2×1)

Step 6: Calculate the result

Now, perform the remaining multiplications and divide them: C(10, 3) = 720 / 6 = 120 There are 120 ways to choose 3 computer chips out of 10.

Key Concepts

Factorials

When we talk about factorials in mathematics, we are referring to the product of a series of descending natural numbers starting from a positive integer and multiplying all the way down to 1. The symbol for a factorial is an exclamation mark (!). For example, if you see the number 5 followed by an exclamation mark, that's denoting '5 factorial', which can be mathematically represented as:

\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials are fundamentally important in the field of combinatorics, where we study the different ways in which objects can be chosen or arranged. They are primarily used in counting problems and in the calculation of permutations and combinations, which are methods to count the possible arrangements or selections of a set of objects without actually listing them.

In the given exercise, the use of factorials simplifies the process of finding out how many ways we can choose 3 computer chips from a set of 10. The factorial notation allows us to compactly represent and easily cancel out the common factors as shown in the step by step solution.

Permutations and Combinations

When it comes to choosing or arranging items, mathematics provides two distinct concepts: permutations and combinations. The key difference is that permutations consider the order to be important, while combinations do not.

A permutation refers to an arrangement of items where the order is taken into account. For example, if we have three books and want to know in how many ways we can arrange them on a shelf, we would calculate a permutation. On the other hand, a combination refers to a selection of items where the order does not matter. If we simply want to know how many ways we can select 2 books out of 4 to take on a trip, we would calculate a combination.

The formulas for permutations and combinations both involve factorials. Specifically, the combination formula used in the exercise is defined as follows:

\( C(n, r) = \frac{n!}{r! \times (n-r)!} \)

Where:\

\

Probability

The concept of probability is concerned with measuring the likelihood of a particular event occurring. In mathematical terms, it is defined as the ratio of the number of favorable outcomes (when the event of interest occurs) to the total number of possible outcomes, assuming each outcome has an equal chance of occurring.

The calculation of probability can often involve combinations. For instance, if each chip in our box of 10 computer chips had a different level of performance, and we wanted to find the probability of choosing a set of 3 specific high-performance chips, we would consider both the number of ways of choosing those 3 chips (the favorable outcome) and the total number of possible combinations of 3 chips from the set of 10.

To calculate such a probability, you would use the following formula:

\( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

Here, the total number of possible outcomes would be the number of combinations of choosing 3 out of 10, which we've already calculated as 120. If, hypothetically, only one particular combination of chips was considered 'high performance', then that would be the only favorable outcome (1 in this case), and the probability P(E) of selecting the high-performance set from the box would simply be \( \frac{1}{120} \) or approximately 0.0083, indicating a very low likelihood.