Question Number 210851 by RojaTaniya last updated on 20/Aug/24
Commented by mr W last updated on 21/Aug/24
$${answer}\:=\mathrm{2024}? \\ $$
Answered by A5T last updated on 21/Aug/24
$$\Rightarrow{b}=\mathrm{4}{k}\Rightarrow\mathrm{3}{a}+\mathrm{21}{k}+\mathrm{7}{c}+\mathrm{21}{d}=\mathrm{506} \\ $$$${a}=\mathrm{7}{q}+\mathrm{3}\Rightarrow\mathrm{21}{q}+\mathrm{21}{k}+\mathrm{7}{c}+\mathrm{21}{d}=\mathrm{497} \\ $$$$\Rightarrow\mathrm{3}{q}+\mathrm{3}{k}+{c}+\mathrm{3}{d}=\mathrm{71}\Rightarrow{c}=\mathrm{3}{m}+\mathrm{2} \\ $$$$\Rightarrow{q}+{k}+{m}+{d}=\mathrm{23};\:{when}\:{q},{k},{m},{d}\geqslant\mathrm{1} \\ $$$${Let}\:{w}={q}−\mathrm{1};{x}={k}−\mathrm{1};{y}={m}−\mathrm{1};{z}={d}−\mathrm{1} \\ $$$$\Rightarrow{w},{x},{y},{z}\geqslant\mathrm{0}\:\wedge\:{w}+{x}+{y}+{z}=\mathrm{19} \\ $$$$\Rightarrow{no}.\:{of}\:{solutions}=\begin{pmatrix}{\mathrm{19}+\mathrm{4}−\mathrm{1}}\\{\mathrm{19}}\end{pmatrix}=\mathrm{1540} \\ $$$${when}\:{q}=\mathrm{0};\:{k}+{m}+{d}=\mathrm{23} \\ $$$${Case}\:{I}:\:{m}=\mathrm{0}\Rightarrow{k}+{d}=\mathrm{23}\:{where}\:{k},{d}\geqslant\mathrm{1} \\ $$$$\Rightarrow{no}.\:{of}\:{solutions}=\mathrm{22} \\ $$$${Case}\:{II}:\:{m}\geqslant\mathrm{1}\Rightarrow{m},{k},{d}\geqslant\mathrm{1} \\ $$$$\Rightarrow{k}−\mathrm{1}+{m}−\mathrm{1}+{d}−\mathrm{1}=\mathrm{20} \\ $$$$\Rightarrow{no}.\:{of}\:{solutions}=\begin{pmatrix}{\mathrm{20}+\mathrm{3}−\mathrm{1}}\\{\mathrm{20}}\end{pmatrix}=\mathrm{231} \\ $$$$\Rightarrow{Total}\:{no}.\:{of}\:{solutions}=\mathrm{1540}+\mathrm{231}+\mathrm{22}=\mathrm{1793} \\ $$