QJEEM24Maf14d: JEE Main 2024 Mathematics - Class 11 JEEM24Maf14d Solution: The number of integers, between 100 and 1000 having the sum of…

Number in this range will be 3-digit number.

$N=\overline{a b c}$ such that $a+b+c=14$

Also, $a \geq 1, \quad a, b, c \in\{0,1,2, \ldots 9\}$

Case I

All 3-digit same

$\Rightarrow 3 a=14$ not possible

Case II

Exactly 2 digit same:

$\Rightarrow 2 a+c=14$

$\begin{aligned}& (a, c) \in\{(3,8),(4,6),(5,4),(6,2),(7,0)\} \\& \Rightarrow\left(\frac{3!}{2!}\right) \text { ways } \Rightarrow 5 \times 3-1 \\& =15-1=14\end{aligned}$

Case III

All digits are distinct

$a+b+c=14$

without losing generality a > b > c

$\begin{aligned}& (a, b, c) \in\left\{\begin{array}{l}(9,5,0),(9,4,1),(9,3,2) \\(8,6,0),(8,5,1),(8,4,2) \\(7,6,1),(7,5,2),(7,4,3) \\(6,5,3)\end{array}\right. \\& \Rightarrow 8 \times 3!+2(3!-2!)=48+8=56 \\& =0+14+56=70\end{aligned}$