Find `sum r^2.^20C_r`

Options

A. `2^20`
B. `2^21`
C. `210xx2^19`
D. `2^19`

Answer

To find the sum \( \sum r^2 \cdot {20 \choose r} \), we can use a known formula for summation involving binomial coefficients. The formula states:

\[ \sum_{r=0}^{n} r^2 \cdot {n \choose r} = n \cdot 2^{n-1} + n(n-1) \cdot 2^{n-2} \]

In our case, \( n = 20 \). Let's apply this formula step by step:

### Step 1: Identify the values
We have \( n = 20 \).

### Step 2: Apply the formula
Using the formula:

\[ \sum_{r=0}^{20} r^2 \cdot {20 \choose r} = 20 \cdot 2^{20-1} + 20(20-1) \cdot 2^{20-2} \]

### Step 3: Simplify the expression
Now, we can simplify each term:

1. The first term:
\[ 20 \cdot 2^{19} \]

2. The second term:
\[ 20 \cdot 19 \cdot 2^{18} \]

### Step 4: Factor out common terms
Now, we can factor out \( 2^{18} \) from both terms:

\[ = 2^{18} \left( 20 \cdot 2 + 20 \cdot 19 \right) \]

### Step 5: Calculate the expression inside the parentheses
Now, calculate \( 20 \cdot 2 + 20 \cdot 19 \):

\[ 20 \cdot 2 = 40 \]
\[ 20 \cdot 19 = 380 \]
\[ 40 + 380 = 420 \]

### Step 6: Write the final expression
Thus, we have:

\[ \sum_{r=0}^{20} r^2 \cdot {20 \choose r} = 2^{18} \cdot 420 \]

### Step 7: Final simplification
We can express \( 420 \) as \( 210 \cdot 2 \):

\[ = 210 \cdot 2^{19} \]

### Conclusion
The final result is:

\[ \sum_{r=0}^{20} r^2 \cdot {20 \choose r} = 210 \cdot 2^{19} \]