To solve the problem, we need to analyze the expressions for \( S_1 \), \( S_2 \), and \( S_3 \) and then combine them to find \( S_1 + S_2 + S_3 \).
### Step 1: Understanding the coefficients
The coefficients \( C_r \) represent the coefficients of \( x^r \) in the binomial expansion of \( (1+x)^{100} \). According to the binomial theorem, we have:
\[
C_r = \binom{100}{r}
\]
for \( r = 0, 1, 2, \ldots, 100 \).
### Step 2: Expressing \( S_1 \), \( S_2 \), and \( S_3 \)
We can express \( S_1 \), \( S_2 \), and \( S_3 \) as follows:
- \( S_1 = \sum_{0 \leq i < j \leq 100} C_i C_j \)
- \( S_2 = \sum_{0 \leq j < i \leq 100} C_i C_j \)
- \( S_3 = \sum_{0 \leq i = j \leq 100} C_i C_j = \sum_{0 \leq i \leq 100} C_i^2 \)
### Step 3: Combining \( S_1 \), \( S_2 \), and \( S_3 \)
Notice that \( S_1 \) and \( S_2 \) can be combined:
\[
S_1 + S_2 = \sum_{0 \leq i \neq j \leq 100} C_i C_j
\]
Thus, we can express the total sum as:
\[
S_1 + S_2 + S_3 = \sum_{0 \leq i, j \leq 100} C_i C_j = \left( \sum_{i=0}^{100} C_i \right)^2
\]
### Step 4: Calculating the total sum
The sum \( \sum_{i=0}^{100} C_i \) is simply:
\[
\sum_{i=0}^{100} C_i = (1 + 1)^{100} = 2^{100}
\]
Thus, we have:
\[
S_1 + S_2 + S_3 = (2^{100})^2 = 2^{200}
\]
### Step 5: Expressing \( S_1 + S_2 + S_3 \) in the form \( a^b \)
We can express \( 2^{200} \) in the form \( a^b \):
\[
2^{200} = (4^{100}) = (16^{50}) = (32^{40})
\]
This gives us multiple representations of \( S_1 + S_2 + S_3 \).
### Step 6: Finding the least value of \( a + b \)
Now we calculate \( a + b \) for each representation:
- For \( 4^{100} \): \( a = 4, b = 100 \) → \( a + b = 104 \)
- For \( 16^{50} \): \( a = 16, b = 50 \) → \( a + b = 66 \)
- For \( 32^{40} \): \( a = 32, b = 40 \) → \( a + b = 72 \)
The least value of \( a + b \) is:
\[
\min(104, 66, 72) = 66
\]
### Final Answer
Thus, the least value of \( a + b \) is \( \boxed{66} \).
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