Find the expansion of `(y+x)^n`

Answer

To find the expansion of \((y + x)^n\) using the Binomial Theorem, we follow these steps:

### Step 1: Understand the Binomial Theorem
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \(\binom{n}{k}\) is the binomial coefficient, which can be calculated as \(\frac{n!}{k!(n-k)!}\).

### Step 2: Identify \(a\) and \(b\)
In our case, we have:
- \(a = y\)
- \(b = x\)

### Step 3: Write the Expansion
Using the Binomial Theorem, we can write the expansion of \((y + x)^n\) as:
\[ (y + x)^n = \sum_{k=0}^{n} \binom{n}{k} y^{n-k} x^k \]

### Step 4: Expand the Summation
Now, we can explicitly write out the terms of the summation:
\[ (y + x)^n = \binom{n}{0} y^n x^0 + \binom{n}{1} y^{n-1} x^1 + \binom{n}{2} y^{n-2} x^2 + \ldots + \binom{n}{n-1} y^1 x^{n-1} + \binom{n}{n} y^0 x^n \]

### Step 5: Write the Final Expansion
Thus, the complete expansion is:
\[ (y + x)^n = \binom{n}{0} y^n + \binom{n}{1} y^{n-1} x + \binom{n}{2} y^{n-2} x^2 + \binom{n}{3} y^{n-3} x^3 + \ldots + \binom{n}{n-1} y x^{n-1} + \binom{n}{n} x^n \]

### Final Result
The expansion of \((y + x)^n\) is:
\[ (y + x)^n = \sum_{k=0}^{n} \binom{n}{k} y^{n-k} x^k \]