In the series, 6 4 1 2 2 8 7 4 2 1 5 3 8 6 2 1 7 1 4 1 3 2 8 6 How many pairs of alternate numbers have a difference of 2?

Options

A. one
B. Two
C. three
D. four

Explanation

To solve the problem of finding how many pairs of alternate numbers in the series have a difference of 2, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Series**: The given series is:
\[ 6, 4, 1, 2, 2, 8, 7, 4, 2, 1, 5, 3, 8, 6, 2, 1, 7, 1, 4, 1, 3, 2, 8, 6 \]

2. **Select Alternate Numbers**: We will take alternate numbers from the series. Starting from the first number, we will skip one number each time:
- First sequence (starting from the first number):
\[ 6, 1, 2, 7, 2, 5, 8, 2, 7, 4, 3, 8 \]
- Second sequence (starting from the second number):
\[ 4, 2, 8, 4, 1, 3, 6, 1, 1, 2, 6 \]

3. **Calculate Differences for the First Sequence**:
- Pairs: (6, 1), (1, 2), (2, 7), (7, 2), (2, 5), (5, 8), (8, 2), (2, 7), (7, 4), (4, 3), (3, 8)
- Differences:
- |6 - 1| = 5
- |1 - 2| = 1
- |2 - 7| = 5
- |7 - 2| = 5
- |2 - 5| = 3
- |5 - 8| = 3
- |8 - 2| = 6
- |2 - 7| = 5
- |7 - 4| = 3
- |4 - 3| = 1
- |3 - 8| = 5

4. **Calculate Differences for the Second Sequence**:
- Pairs: (4, 2), (2, 8), (8, 4), (4, 1), (1, 3), (3, 6), (6, 1), (1, 1), (1, 2), (2, 6)
- Differences:
- |4 - 2| = 2
- |2 - 8| = 6
- |8 - 4| = 4
- |4 - 1| = 3
- |1 - 3| = 2
- |3 - 6| = 3
- |6 - 1| = 5
- |1 - 1| = 0
- |1 - 2| = 1
- |2 - 6| = 4

5. **Count Pairs with a Difference of 2**:
- From the first sequence, there are no pairs with a difference of 2.
- From the second sequence, we have:
- (4, 2) → difference = 2
- (1, 3) → difference = 2
- Therefore, there are **2 pairs** with a difference of 2.

### Final Answer:
The total number of pairs of alternate numbers that have a difference of 2 is **2**.

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