Question

The number of ways in which we can choose $2$ distinct integers from $1to100$ such that difference between them is at most $10$ is

• A. ${}^{40}{C}_2$
• B. ${}^{70}{C}_2$
• C. ${}^{100}{C}_2{-}^{90}{C}_2$
• D. None of these

Answer 1

Answer: (A)

${}^{40}{C}_2$

If you choose any number between 1 and 100 and call it $x$ , you can choose any number between $x+1$ and $x+10$ , which gives us $10$ choices for each.
If you choose $91$ , for example, we have $9$ numbers to choose (i.e. 92 to 100 )
Similarly, $93$ gives us $8$ choices, $94$ gives us $7$ , and so forth, until $99$ gives us only $1$ choice.
Hence, $(90\times 10)+9+8+7++1=945{=}^{10}{C}_2{-}^9{C}_2$