If coefficient of `x^(21)` in the expansion of `(1 + x)^(21) + (1 + x)^(22) + ... + (1 + x)^(30)` is `""^(31)C_(r)`, where `r ge 20`, then r= ___

Answer

To find the value of \( r \) such that the coefficient of \( x^{21} \) in the expansion of \( (1 + x)^{21} + (1 + x)^{22} + \ldots + (1 + x)^{30} \) is equal to \( \binom{31}{r} \) where \( r \geq 20 \), we can follow these steps:

### Step 1: Identify the Coefficient of \( x^{21} \)
The coefficient of \( x^{21} \) in \( (1 + x)^n \) is given by \( \binom{n}{21} \). Therefore, we need to find the coefficients for each term in the series:

- For \( (1 + x)^{21} \): Coefficient of \( x^{21} \) is \( \binom{21}{21} = 1 \)
- For \( (1 + x)^{22} \): Coefficient of \( x^{21} \) is \( \binom{22}{21} = 22 \)
- For \( (1 + x)^{23} \): Coefficient of \( x^{21} \) is \( \binom{23}{21} = 253 \)
- For \( (1 + x)^{24} \): Coefficient of \( x^{21} \) is \( \binom{24}{21} = 2024 \)
- For \( (1 + x)^{25} \): Coefficient of \( x^{21} \) is \( \binom{25}{21} = 12650 \)
- For \( (1 + x)^{26} \): Coefficient of \( x^{21} \) is \( \binom{26}{21} = 65780 \)
- For \( (1 + x)^{27} \): Coefficient of \( x^{21} \) is \( \binom{27}{21} = 296010 \)
- For \( (1 + x)^{28} \): Coefficient of \( x^{21} \) is \( \binom{28}{21} = 982080 \)
- For \( (1 + x)^{29} \): Coefficient of \( x^{21} \) is \( \binom{29}{21} = 3108105 \)
- For \( (1 + x)^{30} \): Coefficient of \( x^{21} \) is \( \binom{30}{21} = 8880300 \)

### Step 2: Sum the Coefficients
Now, we sum all these coefficients:

\[ \text{Total Coefficient} = \binom{21}{21} + \binom{22}{21} + \binom{23}{21} + \binom{24}{21} + \binom{25}{21} + \binom{26}{21} + \binom{27}{21} + \binom{28}{21} + \binom{29}{21} + \binom{30}{21} \]

This can be simplified using the hockey-stick identity in combinatorics, which states that:

\[ \sum_{k=r}^{n} \binom{k}{r} = \binom{n+1}{r+1} \]

In our case, we have:

\[ \sum_{k=21}^{30} \binom{k}{21} = \binom{31}{22} \]

### Step 3: Set the Coefficient Equal to \( \binom{31}{r} \)
From the previous step, we have:

\[ \binom{31}{22} = \binom{31}{r} \]

### Step 4: Solve for \( r \)
Using the property of binomial coefficients, we know that:

\[ \binom{n}{k} = \binom{n}{n-k} \]

Thus, we can set \( r = 22 \) or \( r = 31 - 22 = 9 \). However, since the problem states \( r \geq 20 \), we conclude that:

\[ r = 22 \]

### Final Answer
Therefore, the value of \( r \) is \( \boxed{22} \).