To solve the equation [tex]\({ }^{x-2} C_2 = 21\)[/tex], let's go through the following steps:
1. Understanding the Binomial Coefficient:
The binomial coefficient [tex]\({}^n C_r\)[/tex] is defined as:
[tex]\[
{}^n C_r = \frac{n!}{r!(n-r)!}
\][/tex]
So, [tex]\({ {}^{x-2}} C_2\)[/tex] can be written as:
[tex]\[
{}^{x-2} C_2 = \frac{(x-2)!}{2!(x-2-2)!} = \frac{(x-2)!}{2!(x-4)!}
\][/tex]
Given that this equals 21, we have:
[tex]\[
\frac{(x-2)!}{2!(x-4)!} = 21
\][/tex]
2. Simplifying the Binomial Coefficient Expression:
Notice that we can simplify the factorials:
[tex]\[
\frac{(x-2)!}{2!(x-4)!} = \frac{(x-2)(x-3)(x-4)!}{2!(x-4)!}
\][/tex]
The [tex]\((x-4)!\)[/tex] terms cancel out:
[tex]\[
\frac{(x-2)(x-3)}{2!} = 21 \quad \text{and since} \quad 2! = 2
\][/tex]
This simplifies to:
[tex]\[
\frac{(x-2)(x-3)}{2} = 21
\][/tex]
3. Isolating the Expression:
Multiply both sides by 2 to get rid of the fraction:
[tex]\[
(x-2)(x-3) = 42
\][/tex]
4. Expanding and Forming a Quadratic Equation:
Expand the left-hand side:
[tex]\[
x^2 - 3x - 2x + 6 = 42
\][/tex]
Combine like terms:
[tex]\[
x^2 - 5x + 6 = 42
\][/tex]
5. Solving the Quadratic Equation:
Bring all terms to one side to set the equation to 0:
[tex]\[
x^2 - 5x + 6 - 42 = 0
\][/tex]
Simplify:
[tex]\[
x^2 - 5x - 36 = 0
\][/tex]
Factorizing the quadratic equation:
[tex]\[
(x - 9)(x + 4) = 0
\][/tex]
6. Finding the Solutions:
Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 9 = 0 \quad \Rightarrow \quad x = 9
\][/tex]
[tex]\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\][/tex]
Therefore, the solutions to the equation [tex]\({ }^{x-2} C_2 = 21\)[/tex] are:
[tex]\[
x = 9 \quad \text{and} \quad x = -4
\][/tex]
These values satisfy the original equation.
