Let `t_(100)=sum_(r=0)^(100)(1)/(("^(100)C_(r ))^(5))` and `S_(100)=sum_(r=0)^(100)(r )/(("^(100)C_(r ))^(5))`, then the value of `(100t_(100))/(S_(100))` is (a) 1 (b) 2 (c) 3 (d) 4

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To solve the problem, we need to evaluate the expressions \( T_{100} \) and \( S_{100} \) and find the value of \( \frac{100 T_{100}}{S_{100}} \).

### Step 1: Define the expressions

We have:
\[ T_{100} = \sum_{r=0}^{100} \frac{1}{\left( \binom{100}{r} \right)^5} \]
\[ S_{100} = \sum_{r=0}^{100} \frac{r}{\left( \binom{100}{r} \right)^5} \]

### Step 2: Rewrite \( S_{100} \)

We can rewrite \( S_{100} \) by expressing \( r \) in terms of \( 100 - (100 - r) \):
\[ S_{100} = \sum_{r=0}^{100} \frac{100 - (100 - r)}{\left( \binom{100}{r} \right)^5} \]
This can be separated into two sums:
\[ S_{100} = \sum_{r=0}^{100} \frac{100}{\left( \binom{100}{r} \right)^5} - \sum_{r=0}^{100} \frac{100 - r}{\left( \binom{100}{r} \right)^5} \]

### Step 3: Combine the sums

Notice that the second sum is just \( S_{100} \) again:
\[ S_{100} = 100 T_{100} - S_{100} \]
Adding \( S_{100} \) to both sides gives:
\[ 2 S_{100} = 100 T_{100} \]

### Step 4: Solve for \( \frac{100 T_{100}}{S_{100}} \)

From the equation \( 2 S_{100} = 100 T_{100} \), we can express \( \frac{100 T_{100}}{S_{100}} \):
\[ \frac{100 T_{100}}{S_{100}} = 2 \]

### Conclusion

Thus, the value of \( \frac{100 T_{100}}{S_{100}} \) is:
\[ \boxed{2} \]