Given that (\(_r^n\)) = \(^nC_r\), simplify (\(^{2x + 1}_{3}\)) - (\(^{2x - 1}_3\)) - 2(\(^x_2\))

Given that (\(_r^n\)) = \(^nC_r\), simplify (\(^{2x + 1}_{3}\)) - (\(^{2x - 1}_3\)) - 2(\(^x_2\))

Answer Details

The expression given is: (^2x + 1₃) - (^2x - 1₃) - 2(^x₂)

We can simplify this expression by using the formula (^r_n) = ^nC_r, which represents the number of ways of choosing r objects from a set of n distinct objects.

First, let's simplify (^2x + 1₃) using the formula:

(^2x + 1₃) = ^3C_2x+1

This represents the number of ways of choosing (2x + 1) objects from a set of 3 distinct objects. Similarly, we can simplify (^2x - 1₃) using the formula:

(^2x - 1₃) = ^3C_2x-1

This represents the number of ways of choosing (2x - 1) objects from a set of 3 distinct objects.

Now, substituting these values in the given expression, we get:

(^2x + 1₃) - (^2x - 1₃) - 2(^x₂) = ^3C_2x+1 - ^3C_2x-1 - 2(^x₂)

Next, we can simplify 2(^x₂) using the formula:

2(^x₂) = 2^xC₂

This represents the number of ways of choosing 2 objects from a set of x distinct objects, multiplied by 2.

Substituting this value in the previous expression, we get:

(^2x + 1₃) - (^2x - 1₃) - 2(^x₂) = ^3C_2x+1 - ^3C_2x-1 - 2^xC₂

This is the simplified expression, which represents the difference between the number of ways of choosing (2x + 1) objects and (2x - 1) objects from a set of 3 distinct objects, minus twice the number of ways of choosing 2 objects from a set of x distinct objects.