Mathematics - Binomial Theorem Question with Solution | TestHub

$\mathrm{N}=\mathrm{ab}\mathrm{c}$ (i) $\begin{array}{rl}& \text{ All distinct digits }\\ & \mathrm{a}+\mathrm{b}+\mathrm{c}=14\\ & \mathrm{a}\ge 1\\ & \mathrm{b},\mathrm{c}\in \{0\text{ to }9\}\end{array}$ $\begin{array}{lll}byhit\&trial:& 8cases& \\ (6,5,3)& (8,6,0)& (9,4,1)\\ (7,6,1)& (8,5,1)& (9,3,2)\\ (7,5,2)& (8,4,2)& \\ (7,4,3)& (9,5,0)& \end{array}$ (ii) 2 same, 1 diff $a=b$;c $2\mathrm{a}+\mathrm{c}=14$ by values : $\begin{array}{rl}& \begin{array}{l}(3,8)\\ (4,6)\\ (5,4)\\ (6,2)\\ (7,0)\end{array}\}\frac{3!}{2!}\times 5-1\\ & =14\text{ cases }\end{array}$ (iii) all same : $\begin{array}{rl}& \begin{array}{l}3a=14\\ a=\frac{14}{3}\times \text{ rejected }\\ 0\text{ cases }\end{array}\end{array}$ Hence, Total cases: $\begin{array}{rl}& 8\times 3!+2\times (4)+14\\ & =48+22\\ & =70\end{array}$