For a binomial experiment, show that the probability of  successes  is given by:
 

In a binomial experiment, we deal with a situation where each trial has two possible outcomes: success or failure. A typical example is flipping a coin where the outcomes can be "heads" (success) or "tails" (failure). The binomial experiment is characterized by the following properties:

1. There are \( n \) independent trials.
2. Each trial has two possible outcomes: success (denoted as \( x \)) or failure (denoted as \( n - x \)).
3. The probability of success on any given trial is \( p \), and the probability of failure is \( 1 - p \).
4. The number of successes, \( x \), in \( n \) trials is of interest.

### Step 1: Understanding the Problem

We are interested in finding the probability of having exactly \( x \) successes in a binomial experiment. The total number of trials is \( n \), and the probability of success in each trial is \( p \), while the probability of failure is \( 1 - p \).

### Step 2: Defining the Binomial Probability

We need to compute the probability that \( x \) successes occur in \( n \) trials. The probability of exactly \( x \) successes can be derived from the following steps:

- In each trial, there are two possible outcomes: success or failure.
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