Problem 51 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Field Trip In how many ways can seven students from a class of 30 be chosen for a field trip?

Step-by-Step Solution

Answer

There are 2,035,800 ways to choose seven students from a class of 30.

Step 1: Identify the Problem Type

The problem involves choosing a subset of students from a larger set without regard to order. This is a typical combination problem in mathematics, where we need to find how many ways a subset can be selected from a larger set.

Step 2: Understand the Formula

The formula for combinations, given by \( nCr \), is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n \) represents the total number of items to choose from, and \( r \) represents the number of items to choose.

Step 3: Substitute Values

Substitute \( n = 30 \) and \( r = 7 \) into the combination formula. This gives us \( \binom{30}{7} = \frac{30!}{7!(30-7)!} \).

Step 4: Calculate the Factorials

Calculate the factorials: \( 30! \) is a large number, so we focus on simplifying the division with \( \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25 \times 24}{7!} \).

Step 5: Simplify the Fraction

Calculate \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \). Substitute this into the combination formula to get the fraction: \( \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25 \times 24}{5040} \).

Step 6: Compute the Result

Perform the calculations to obtain the final result. Dividing the product by 5040 gives us 2,035,800 ways.

Key Concepts

Factorial

Factorial is a mathematical operation used to calculate the product of an integer and all the positive integers below it down to 1. It's represented by the exclamation mark symbol "!". For instance, the factorial of 5, written as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials grow very quickly as the number increases. This is why with large numbers, like 30!, calculations would become quite cumbersome if done entirely by hand. However, in combination problems, we often need only a portion of these calculations, which can often be canceled out when dividing, making it more manageable. This simplification is crucial in computing combinations efficiently.

Combination Formula

The combination formula is a mathematical expression used to determine how many different subsets of a certain size can be selected from a larger set, without regard to the order of selection. This formula is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]Here, \(n\) is the total number of items, while \(r\) is the number of items to be chosen. This formula helps in figuring out how many ways a subset of size \(r\) can be picked from a set of \(n\) items.

• The top part, \(n!\), accounts for all possible arrangements of \(n\) items.
• The bottom part, \(r!(n-r)!\), adjusts this count for duplicities, dividing by the number of arrangements within the subsets and unselected items.

This formula is central in determining the number of ways to choose \(r\) out of \(n\) without care for order.

Subsets

Subsets are defined as any grouping of elements from a set, including the empty set and the set itself. For example, if you have a set \(S\) consisting of the elements \( \{a, b, c\}\), the subsets of \(S\) include \( \{\}, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\} \).
In combination problems like the one involving choosing students for a field trip, we are interested in subsets of a certain size, specifically subsets of 7 students out of the larger set of 30 students.

• These subsets are all possible ways to choose exactly 7 students.
• Order does not matter in these subsets, meaning the group \( \{1, 2, 3, 4, 5, 6, 7\} \) is the same as \( \{7, 6, 5, 4, 3, 2, 1\} \).
• Thus, the combination formula helps us compute the total number of such subsets possible without listing them exhaustively.

nCr

In mathematics, \(nCr\) is shorthand notation for the combination formula. It is pronounced "n choose r". \(nCr\) specifically calculates the number of ways to choose \(r\) elements from a total \(n\) elements. This expression does so without considering the order of selection. For example, in the problem of selecting 7 students from a class of 30, we use the expression \(\binom{30}{7}\), which translates into:

• \(n = 30\) – the total number of students.
• \(r = 7\) – the number of students to be chosen for the trip.
• Using this, \(\binom{30}{7} = \frac{30!}{7! \, (30-7)!}\).

The computed value \(2035800\) reflects the total number of unique ways in which 7 students can be chosen from 30. This method efficiently bypasses listing out every possible combination, using mathematical permutation concepts instead.