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Question-210851




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

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