The number of ways in which we can choose 2 distinct integers from 1 to 100 such that difference between them is at most 10 is

Options

A. `""^(40)C_(2)`
B. `""^(70)C_(2)`
C. `""^(100)C_(2)-""^(90)C_(2)`
D. 945

Answer

To solve the problem of choosing 2 distinct integers from 1 to 100 such that the difference between them is at most 10, we can break down the solution step by step.

### Step 1: Understanding the Range
We are tasked with selecting two distinct integers \( a \) and \( b \) such that \( 1 \leq a < b \leq 100 \) and \( b - a \leq 10 \).

### Step 2: Setting Up the Choices
For each integer \( a \) from 1 to 90, we can find the corresponding values of \( b \) that satisfy the condition \( b - a \leq 10 \).

### Step 3: Counting Valid Choices for Each \( a \)
1. **For \( a = 1 \)**:
- Possible values for \( b \): 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 (10 choices)

2. **For \( a = 2 \)**:
- Possible values for \( b \): 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (10 choices)

3. **For \( a = 3 \)**:
- Possible values for \( b \): 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (10 choices)

Continuing this pattern, we see that for every \( a \) from 1 to 90, we can always choose 10 different values for \( b \).

### Step 4: Special Cases for Higher Values of \( a \)
- **For \( a = 91 \)**:
- Possible values for \( b \): 92, 93, 94, 95, 96, 97, 98, 99, 100 (9 choices)

- **For \( a = 92 \)**:
- Possible values for \( b \): 93, 94, 95, 96, 97, 98, 99, 100 (8 choices)

- **For \( a = 93 \)**:
- Possible values for \( b \): 94, 95, 96, 97, 98, 99, 100 (7 choices)

- **For \( a = 94 \)**:
- Possible values for \( b \): 95, 96, 97, 98, 99, 100 (6 choices)

- **For \( a = 95 \)**:
- Possible values for \( b \): 96, 97, 98, 99, 100 (5 choices)

- **For \( a = 96 \)**:
- Possible values for \( b \): 97, 98, 99, 100 (4 choices)

- **For \( a = 97 \)**:
- Possible values for \( b \): 98, 99, 100 (3 choices)

- **For \( a = 98 \)**:
- Possible values for \( b \): 99, 100 (2 choices)

- **For \( a = 99 \)**:
- Possible value for \( b \): 100 (1 choice)

### Step 5: Summing Up the Choices
Now, we can sum the total number of choices:
- From \( a = 1 \) to \( a = 90 \): \( 90 \times 10 = 900 \)
- From \( a = 91 \) to \( a = 99 \): \( 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 \)

### Final Calculation
Total number of ways = \( 900 + 45 = 945 \)

Thus, the number of ways in which we can choose 2 distinct integers from 1 to 100 such that the difference between them is at most 10 is **945**.