Combination Worksheets - Free Printable

The image contains a worksheet titled "Evaluating Combinations," which involves solving problems related to combinations (denoted as \( C(n, r) \) or \( \binom{n}{r} \)). The formula for combinations is:

\[
C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \).

Let's solve a few of the problems step by step to illustrate the process. I'll choose some representative examples from the worksheet.

---

Problem 1: \( 4C(14, 2) \)

#### Step 1: Understand the problem
We need to calculate \( C(14, 2) \) and then multiply the result by 4.

#### Step 2: Use the combination formula
\[
C(14, 2) = \binom{14}{2} = \frac{14!}{2!(14-2)!} = \frac{14!}{2! \cdot 12!}
\]

#### Step 3: Simplify the factorials
\[
\frac{14!}{2! \cdot 12!} = \frac{14 \times 13 \times 12!}{2! \cdot 12!} = \frac{14 \times 13}{2!}
\]
Since \( 2! = 2 \times 1 = 2 \):
\[
\frac{14 \times 13}{2} = \frac{182}{2} = 91
\]

#### Step 4: Multiply by 4
\[
4 \times 91 = 364
\]

#### Final Answer:
\[
\boxed{364}
\]

---

Problem 2: \( C(30, 20) - C(30, 15) \)

#### Step 1: Understand the problem
We need to calculate \( C(30, 20) \) and \( C(30, 15) \), then subtract the latter from the former.

#### Step 2: Use the combination formula for \( C(30, 20) \)
\[
C(30, 20) = \binom{30}{20} = \frac{30!}{20!(30-20)!} = \frac{30!}{20! \cdot 10!}
\]

#### Step 3: Simplify the factorials
\[
\frac{30!}{20! \cdot 10!} = \frac{30 \times 29 \times 28 \times \cdots \times 21 \times 20!}{20! \cdot 10!} = \frac{30 \times 29 \times 28 \times \cdots \times 21}{10!}
\]
Calculating \( 10! \):
\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]
So:
\[
C(30, 20) = \frac{30 \times 29 \times 28 \times \cdots \times 21}{3,628,800}
\]
This can be computed using a calculator or software, and the result is:
\[
C(30, 20) = 30,045,015
\]

#### Step 4: Use the combination formula for \( C(30, 15) \)
\[
C(30, 15) = \binom{30}{15} = \frac{30!}{15!(30-15)!} = \frac{30!}{15! \cdot 15!}
\]

#### Step 5: Simplify the factorials
\[
\frac{30!}{15! \cdot 15!} = \frac{30 \times 29 \times 28 \times \cdots \times 16 \times 15!}{15! \cdot 15!} = \frac{30 \times 29 \times 28 \times \cdots \times 16}{15!}
\]
Calculating \( 15! \):
\[
15! = 1,307,674,368,000
\]
So:
\[
C(30, 15) = \frac{30 \times 29 \times 28 \times \cdots \times 16}{1,307,674,368,000}
\]
This can be computed using a calculator or software, and the result is:
\[
C(30, 15) = 155,117,520
\]

#### Step 6: Subtract the two results
\[
C(30, 20) - C(30, 15) = 30,045,015 - 155,117,520 = -125,072,505
\]

#### Final Answer:
\[
\boxed{-125,072,505}
\]

---

Problem 3: \( \frac{C(20, 5)}{C(20, 4)} \)

#### Step 1: Understand the problem
We need to calculate \( C(20, 5) \) and \( C(20, 4) \), then divide the former by the latter.

#### Step 2: Use the combination formula for \( C(20, 5) \)
\[
C(20, 5) = \binom{20}{5} = \frac{20!}{5!(20-5)!} = \frac{20!}{5! \cdot 15!}
\]

#### Step 3: Simplify the factorials
\[
\frac{20!}{5! \cdot 15!} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15!}{5! \cdot 15!} = \frac{20 \times 19 \times 18 \times 17 \times 16}{5!}
\]
Calculating \( 5! \):
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]
So:
\[
C(20, 5) = \frac{20 \times 19 \times 18 \times 17 \times 16}{120}
\]
This can be computed using a calculator or software, and the result is:
\[
C(20, 5) = 15,504
\]

#### Step 4: Use the combination formula for \( C(20, 4) \)
\[
C(20, 4) = \binom{20}{4} = \frac{20!}{4!(20-4)!} = \frac{20!}{4! \cdot 16!}
\]

#### Step 5: Simplify the factorials
\[
\frac{20!}{4! \cdot 16!} = \frac{20 \times 19 \times 18 \times 17 \times 16!}{4! \cdot 16!} = \frac{20 \times 19 \times 18 \times 17}{4!}
\]
Calculating \( 4! \):
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
So:
\[
C(20, 4) = \frac{20 \times 19 \times 18 \times 17}{24}
\]
This can be computed using a calculator or software, and the result is:
\[
C(20, 4) = 4,845
\]

#### Step 6: Divide the two results
\[
\frac{C(20, 5)}{C(20, 4)} = \frac{15,504}{4,845} = 3.2
\]

#### Final Answer:
\[
\boxed{3.2}
\]

---

General Approach

1. Identify the values of \( n \) and \( r \) in each combination.
2. Use the formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
3. Simplify the factorials by canceling out common terms.
4. Compute the result using a calculator if necessary.
5. Perform any additional operations (e.g., addition, subtraction, multiplication, division) as required by the problem.

If you need solutions for more specific problems from the worksheet, please let me know!