If `Sigma_(i=1)^(20) ((""^(20)C_(i-1))/(""^(20)C_(i)+""^(20)C_(i-1)))^(3)=(k)/(21)`, then k equals

Options

A. 200
B. 50
C. 100
D. 400

Answer

To solve the problem, we need to evaluate the expression given in the summation and find the value of \( k \) such that:

\[ \sum_{i=1}^{20} \left( \frac{\binom{20}{i-1}}{\binom{20}{i} + \binom{20}{i-1}} \right)^3 = \frac{k}{21} \]

### Step 1: Simplify the Fraction Inside the Summation
Using the identity for binomial coefficients:

\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]

we can rewrite the denominator:

\[ \binom{20}{i} + \binom{20}{i-1} = \binom{21}{i} \]

Thus, we can express the fraction as:

\[ \frac{\binom{20}{i-1}}{\binom{21}{i}} \]

### Step 2: Rewrite the Summation
Now, substituting this back into the summation gives us:

\[ \sum_{i=1}^{20} \left( \frac{\binom{20}{i-1}}{\binom{21}{i}} \right)^3 \]

### Step 3: Express the Binomial Coefficient
We can express the binomial coefficient as:

\[ \frac{\binom{20}{i-1}}{\binom{21}{i}} = \frac{\binom{20}{i-1}}{\frac{21}{i} \binom{20}{i-1}} = \frac{i}{21} \]

### Step 4: Substitute Back into the Summation
Now substituting this back into the summation, we have:

\[ \sum_{i=1}^{20} \left( \frac{i}{21} \right)^3 = \frac{1}{21^3} \sum_{i=1}^{20} i^3 \]

### Step 5: Calculate the Sum of Cubes
The formula for the sum of the first \( n \) cubes is:

\[ \sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2 \]

For \( n = 20 \):

\[ \sum_{i=1}^{20} i^3 = \left( \frac{20 \cdot 21}{2} \right)^2 = (210)^2 = 44100 \]

### Step 6: Substitute Back to Find the Summation
Now substituting this back into our expression:

\[ \sum_{i=1}^{20} \left( \frac{i}{21} \right)^3 = \frac{1}{21^3} \cdot 44100 \]

### Step 7: Calculate the Final Value
Calculating \( \frac{44100}{21^3} \):

\[ 21^3 = 9261 \]

So we have:

\[ \frac{44100}{9261} \]

### Step 8: Simplify the Result
Now we simplify:

\[ \frac{44100 \div 441}{9261 \div 441} = \frac{100}{21} \]

### Step 9: Relate to k
From our original equation, we have:

\[ \frac{100}{21} = \frac{k}{21} \]

Thus, we find:

\[ k = 100 \]

### Final Answer
The value of \( k \) is:

\[ \boxed{100} \]