https://r2.erweima.ai/mathbot/image/1724952778879lrx6vprh.jpg

A sentença   é válida se e somente se n! for igual a:

The expression given is:

\[
\binom{n+2}{n} = 10
\]

This is a binomial coefficient, which is calculated using the formula:

\[
\binom{n+2}{n} = \frac{(n+2)!}{n!(n+2-n)!} = \frac{(n+2)!}{n! \cdot 2!}
\]

Given that this equals 10, we can set up the equation:

\[
\frac{(n+2)!}{n! \cdot 2!} = 10
\]

Simplifying further:

\[
\frac{(n+2)(n+1)n!}{n! \cdot 2} = 10
\]

Canceling out \(n!\) from both sides:

\[
\frac{(n+2)(n+1)}{2} = 10
\]

Multiply both sides by 2:

\[
(n+2)(n+1) = 20
\]

Now, let's find the value of \(n\) by solving the quadratic equation:

\[
n^2 + 3n + 2 = 20
\]

Subtract 20 from both sides:

\[
n^2 + 3n - 18 = 0
\]

To solve this quadratic equation, we factor it:

\[
(n + 6)(n - 3) = 0
\]

This gives us two possible solutions for \(n\):

\[
n = -6 \quad \text{or} \quad n = 3
\]

Since \(n\) represents a number of elements, \(n\) must be non-negative. Therefore, \(n = 3\).

Finally, let's calculate \(n!\) to satisfy the condition:

\[
n! = 3! = 3 \times 2 \times 1 = 6
\]

Thus, the value of \(n!\) is \(6\).

Would you like more details, or do you have any questions? Here are five related questions that might interest you:

1. What is the general formula for the binomial coefficient \(\binom{n}{k}\)?
2. How do you solve a quadratic equation using the quadratic formula?
3. What are the properties of factorials?
4. Can you find other values of \(n\) for different binomial coefficients that equal 10?
5. How do we handle negative roots in quadratic equations?

**Tip:** When dealing with factorials and binomial coefficients, always check for simplifications that can make solving the equation easier.

Learn how to solve the equation involving the binomial coefficient and factorial, where n! must equal the given number. This problem covers key concepts in combinatorics, including the binomial coefficient formula and factorial properties. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation