Find the coefficient of `x^(17) `in `(x+y)^20`

Answer

To find the coefficient of \( x^{17} \) in \( (x+y)^{20} \), we can use the Binomial Theorem, which states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

In our case, we have \( a = x \), \( b = y \), and \( n = 20 \). We want to find the coefficient of \( x^{17} \).

### Step 1: Identify the values
From the expression \( (x+y)^{20} \):
- \( n = 20 \)
- We want \( x^{17} \), which means \( r \) (the exponent of \( y \)) will be \( n - r = 17 \) or \( r = 20 - 17 = 3 \).

### Step 2: Use the Binomial Coefficient
The coefficient of \( x^{17} \) in \( (x+y)^{20} \) can be found using the binomial coefficient:

\[ \text{Coefficient} = \binom{20}{3} \cdot x^{17} \cdot y^3 \]

### Step 3: Calculate the Binomial Coefficient
Now we need to calculate \( \binom{20}{3} \):

\[ \binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20!}{3! \cdot 17!} \]

Calculating this gives:

\[ \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = \frac{6840}{6} = 1140 \]

### Step 4: Write the Final Coefficient
Thus, the coefficient of \( x^{17} \) in \( (x+y)^{20} \) is:

\[ \text{Coefficient} = 1140 \]

### Final Answer
The coefficient of \( x^{17} \) in \( (x+y)^{20} \) is **1140**.

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