The number of ways in which we can choose 2 distinct integers from 1 to 200 so that the differecne between them is atmost 20 is

Options

A. 3790
B. `.^(200)C_(2)-.^(180)C_(2)`
C. `.^(180)C_(1)xx20+(19xx20)/2`
D. `.^(180)C_(2)`

Answer

To solve the problem of choosing 2 distinct integers from 1 to 200 such that the difference between them is at most 20, we can follow these steps:

### Step 1: Understand the Range of Integers
We are selecting integers from the set {1, 2, ..., 200}. The maximum difference allowed between the two integers is 20.

### Step 2: Set Up the Problem
Let’s denote the two distinct integers as \( x \) and \( y \) where \( x < y \). The condition \( y - x \leq 20 \) implies that \( y \) can be at most \( x + 20 \).

### Step 3: Determine the Possible Values for \( x \)
The smallest value for \( x \) is 1. If \( x = 1 \), then \( y \) can take values from 2 to 21 (total of 20 options).

As \( x \) increases, the maximum value for \( y \) also changes:
- If \( x = 2 \), \( y \) can be from 3 to 22 (20 options).
- If \( x = 3 \), \( y \) can be from 4 to 23 (20 options).
- Continue this pattern until \( x = 180 \), where \( y \) can be from 181 to 200 (20 options).
- If \( x = 181 \), \( y \) can be from 182 to 200 (19 options).
- If \( x = 182 \), \( y \) can be from 183 to 200 (18 options).
- Continue this until \( x = 199 \), where \( y \) can only be 200 (1 option).

### Step 4: Count the Total Combinations
Now, we can summarize the number of valid pairs \( (x, y) \):
- For \( x = 1 \) to \( 180 \): 20 options each (180 * 20 = 3600)
- For \( x = 181 \): 19 options
- For \( x = 182 \): 18 options
- For \( x = 183 \): 17 options
- For \( x = 184 \): 16 options
- For \( x = 185 \): 15 options
- For \( x = 186 \): 14 options
- For \( x = 187 \): 13 options
- For \( x = 188 \): 12 options
- For \( x = 189 \): 11 options
- For \( x = 190 \): 10 options
- For \( x = 191 \): 9 options
- For \( x = 192 \): 8 options
- For \( x = 193 \): 7 options
- For \( x = 194 \): 6 options
- For \( x = 195 \): 5 options
- For \( x = 196 \): 4 options
- For \( x = 197 \): 3 options
- For \( x = 198 \): 2 options
- For \( x = 199 \): 1 option

### Step 5: Calculate the Total
Now, we can calculate the total number of valid pairs:
\[ \text{Total} = 3600 + 19 + 18 + 17 + 16 + 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 \]
The sum of the last 19 natural numbers (from 1 to 19) can be calculated using the formula for the sum of the first \( n \) natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} = \frac{19 \times 20}{2} = 190 \]
So, the total becomes:
\[ \text{Total} = 3600 + 190 = 3790 \]

### Final Answer
The number of ways to choose 2 distinct integers from 1 to 200 such that their difference is at most 20 is **3790**.

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