Coefficient of `x^5` in `( 1+ x)^21 + (1 + x)^22 + …….+ (1 + x)^30` is

Options

A. `""^51C_5`
B. `""^9C_5`
C. `""^31C_6 - ""^21C_6`
D. `""^30C_5 + ""^20C_5`

Answer

To find the coefficient of \( x^5 \) in the expression \( (1+x)^{21} + (1+x)^{22} + \ldots + (1+x)^{30} \), we can follow these steps:

### Step 1: Define the expression
Let
\[ S = (1+x)^{21} + (1+x)^{22} + (1+x)^{23} + (1+x)^{24} + (1+x)^{25} + (1+x)^{26} + (1+x)^{27} + (1+x)^{28} + (1+x)^{29} + (1+x)^{30} \]

### Step 2: Use the property of binomial coefficients
The coefficient of \( x^r \) in \( (1+x)^n \) is given by \( \binom{n}{r} \). Therefore, the coefficient of \( x^5 \) in \( S \) can be expressed as:
\[ \text{Coefficient of } x^5 \text{ in } S = \binom{21}{5} + \binom{22}{5} + \binom{23}{5} + \binom{24}{5} + \binom{25}{5} + \binom{26}{5} + \binom{27}{5} + \binom{28}{5} + \binom{29}{5} + \binom{30}{5} \]

### Step 3: Calculate the coefficients
Now we will calculate each of these binomial coefficients:
- \( \binom{21}{5} = \frac{21 \times 20 \times 19 \times 18 \times 17}{5 \times 4 \times 3 \times 2 \times 1} = 20349 \)
- \( \binom{22}{5} = \frac{22 \times 21 \times 20 \times 19 \times 18}{5 \times 4 \times 3 \times 2 \times 1} = 26334 \)
- \( \binom{23}{5} = \frac{23 \times 22 \times 21 \times 20 \times 19}{5 \times 4 \times 3 \times 2 \times 1} = 33649 \)
- \( \binom{24}{5} = \frac{24 \times 23 \times 22 \times 21 \times 20}{5 \times 4 \times 3 \times 2 \times 1} = 42504 \)
- \( \binom{25}{5} = \frac{25 \times 24 \times 23 \times 22 \times 21}{5 \times 4 \times 3 \times 2 \times 1} = 53130 \)
- \( \binom{26}{5} = \frac{26 \times 25 \times 24 \times 23 \times 22}{5 \times 4 \times 3 \times 2 \times 1} = 65780 \)
- \( \binom{27}{5} = \frac{27 \times 26 \times 25 \times 24 \times 23}{5 \times 4 \times 3 \times 2 \times 1} = 80730 \)
- \( \binom{28}{5} = \frac{28 \times 27 \times 26 \times 25 \times 24}{5 \times 4 \times 3 \times 2 \times 1} = 98280 \)
- \( \binom{29}{5} = \frac{29 \times 28 \times 27 \times 26 \times 25}{5 \times 4 \times 3 \times 2 \times 1} = 118755 \)
- \( \binom{30}{5} = \frac{30 \times 29 \times 28 \times 27 \times 26}{5 \times 4 \times 3 \times 2 \times 1} = 142506 \)

### Step 4: Sum the coefficients
Now we sum all these coefficients:
\[ \text{Total} = 20349 + 26334 + 33649 + 42504 + 53130 + 65780 + 80730 + 98280 + 118755 + 142506 \]
Calculating this gives:
\[ \text{Total} = 20349 + 26334 + 33649 + 42504 + 53130 + 65780 + 80730 + 98280 + 118755 + 142506 = 1040067 \]

### Final Answer
Thus, the coefficient of \( x^5 \) in \( S \) is \( 1040067 \).

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