The number of ways in which we can choose two distinct integers from one to hundred such that difference between them is at most 10 is

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$Number of way choosing the distict integer from 1 to 100 such that difference between then is at most 10 (inclusive) =# |(x, y) \in ℕ \times ℕ : x \neq y, |x - y| ⩽ 10 and 1 ⩽ x, y ⩽ 100}| = \frac{1}{2} {[number of tumple choosen for (x, y) with givne property for 1 ⩽ x ⩽ 10] + [number of tumple choosen for (x, y) with givne property for 91 ⩽ x ⩽ 100] + [number of tumple choosen for (x, y) with givne property for 11 ⩽ x ⩽ 90]} = \frac{1}{2} {(10 + 11 + 12 + . . 19) + (10 + 11 + 12 + . . . + 19) + 20 \times 90} = \frac{1}{2} (2090) = 1045$

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