Problem 41 - Algebra and Trigonometry

Question

In how many ways can a committee of 4 students be formed from a pool of 7 students?

Step-by-Step Solution

Answer

35

Step 1: Understand the Problem

Determine the number of possible ways to select a committee of 4 students from a pool of 7 students. This is a combination problem because the order of selection does not matter.

Step 2: Identify the Formula for Combinations

Use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of students, and \( r \) is the number of students to be chosen for the committee.

Step 3: Substitute the Given Values into the Formula

For this problem, \( n = 7 \) and \( r = 4 \). Substitute these values into the combination formula: \[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \times 3!} \]

Step 4: Calculate the Factorials

Calculate the factorials: \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \] \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] \[ 3! = 3 \times 2 \times 1 = 6 \]

Step 5: Compute the Combination

Substitute the factorials back into the combination formula: \[ \binom{7}{4} = \frac{5040}{24 \times 6} = \frac{5040}{144} = 35 \]

Key Concepts

factorials

Factorials are a fundamental concept in mathematics, especially in combinatorics. A factorial is the product of all positive integers up to a given number. It's denoted by an exclamation mark, for example, 7 factorial is written as 7!. It's calculated as follows:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.
Let's break this down:

• The factorial of 0 is defined as 1 (0! = 1).
• The factorial of 1 is 1 (1! = 1).
• For any number n greater than 1, n! is the product of all integers from 1 to n.

Factorials grow very quickly with larger numbers, making them essential for calculations in permutations and combinations.

combination formula

When we talk about combinations, we refer to selecting items from a larger pool, where the order does not matter. The combination formula is:
\( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
Here:

• n is the total number of items.
• r is the number of items to choose.
• n! is the factorial of n.
• r! is the factorial of r.
• (n-r)! is the factorial of the difference between n and r.

For example, if we want to choose 4 students out of 7, we calculate:
\( \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \times 3!} \). This value tells us the number of ways to pick 4 students from a group of 7.

problem-solving steps

Understanding how to approach combination problems step by step is very useful. Let's use the example of forming a committee of 4 from 7 students to illustrate:

• Step 1: Identify that the problem is a combination, not a permutation since the order does not matter.
• Step 2: Write down the combination formula:

\( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

• Step 3: Substitute the values (n = 7, r = 4):

\( \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \times 3!} \)

• Step 4: Calculate the factorials:

7! = 5040, 4! = 24, and 3! = 6.

• Step 5: Substitute back into the formula:

\( \binom{7}{4} = \frac{5040}{24 \times 6} = \frac{5040}{144} = 35 \)
This means there are 35 different ways to form the committee. Following these steps ensures you understand each part of the problem and leads you to the correct solution.