The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________
The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________
Solution
$\mathrm{N}=\mathrm{ab} \mathrm{c}$ (i)
$\begin{aligned}
& \text { All distinct digits } \\
& \mathrm{a}+\mathrm{b}+\mathrm{c}=14 \\
& \mathrm{a} \geq 1 \\
& \mathrm{~b}, \mathrm{c} \in\{0 \text { to } 9\}
\end{aligned}$
$\begin{array}{lll}
by hit \& trial : & 8 cases \\
(6,5,3) & (8,6,0) & (9,4,1) \\
(7,6,1) & (8,5,1) & (9,3,2) \\
(7,5,2) & (8,4,2) & \\
(7,4,3) & (9,5,0) &
\end{array}$
(ii) 2 same, 1 diff $\quad a=b$;c
$2 \mathrm{a}+\mathrm{c}=14$
by values :
$\begin{aligned}
& \left.\begin{array}{l}
(3,8) \\
(4,6) \\
(5,4) \\
(6,2) \\
(7,0)
\end{array}\right\} \frac{3 !}{2 !} \times 5-1 \\
& =14 \text { cases }
\end{aligned}$
(iii) all same :
$\begin{aligned}
& \begin{array}{l}
3 a=14 \\
a=\frac{14}{3} \times \text { rejected } \\
0 \text { cases }
\end{array}
\end{aligned}$
Hence, Total cases:
$\begin{aligned}
& 8 \times 3 !+2 \times(4)+14 \\
& =48+22 \\
& =70
\end{aligned}$
Asked in: JEE Main 2024 (09 Apr Shift 2)
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