If the digits at ten's and hundred's places in `(11)^(2016)` are x and y respectively, then the order pair (x, y) is equal to:

Options

A. (1, 8)
B. (1, 6)
C. (6, 1)
D. (8, 1)

Answer

To find the digits at the ten's and hundred's places in \( (11)^{2016} \), we can use the Binomial Theorem. Let's break down the solution step by step:

### Step 1: Apply the Binomial Theorem
Using the Binomial Theorem, we can express \( (11)^{2016} \) as:
\[ (10 + 1)^{2016} \]
According to the Binomial Theorem, this expands to:
\[ \sum_{k=0}^{2016} \binom{2016}{k} 10^k \cdot 1^{2016-k} = \sum_{k=0}^{2016} \binom{2016}{k} 10^k \]

### Step 2: Identify Relevant Terms
We are interested in the digits at the ten's and hundred's places, which correspond to the coefficients of \( 10^1 \) and \( 10^2 \) in the expansion. Thus, we need to find:
- The coefficient of \( 10^1 \) (which gives us the ten's place digit)
- The coefficient of \( 10^2 \) (which gives us the hundred's place digit)

### Step 3: Calculate the Coefficients
The coefficient of \( 10^1 \) is given by \( \binom{2016}{1} \) and the coefficient of \( 10^2 \) is given by \( \binom{2016}{2} \).

Calculating these:
- \( \binom{2016}{1} = 2016 \)
- \( \binom{2016}{2} = \frac{2016 \times 2015}{2} = 1008 \times 2015 \)

### Step 4: Find the Digits
Now we need to extract the last two digits from these coefficients to find the ten's and hundred's places.

1. The last digit of \( 2016 \) is \( 6 \).
2. To find the last two digits of \( 1008 \times 2015 \):
- Calculate \( 1008 \times 2015 \):
\[ 1008 \times 2015 = 1008 \times (2000 + 15) = 1008 \times 2000 + 1008 \times 15 \]
- \( 1008 \times 2000 = 2016000 \) (last two digits are \( 00 \))
- \( 1008 \times 15 = 15120 \) (last two digits are \( 20 \))
- Adding these gives \( 00 + 20 = 20 \).

Thus, the last two digits of \( 1008 \times 2015 \) are \( 20 \).

### Conclusion
From our calculations:
- The ten's place (x) is \( 6 \).
- The hundred's place (y) is \( 2 \).

Thus, the ordered pair \( (x, y) \) is \( (6, 2) \).

### Final Answer
The ordered pair \( (x, y) \) is \( (6, 2) \).

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