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Question Number 56913 by Joel578 last updated on 26/Mar/19

How many possible solution sets that satisfy x_1 + x_2 + x_3 + x_4 = 5 with 0 ≤ x_1 ≤ 3 0 ≤ x_2 ≤ 3 0 ≤ x_3 ≤ 2 0 ≤ x_4 ≤ 2

$$\mathrm{How}\:\mathrm{many}\:\mathrm{possible}\:\mathrm{solution}\:\mathrm{sets}\:\mathrm{that}\:\mathrm{satisfy}\: \\ $$$${x}_{\mathrm{1}} \:+\:{x}_{\mathrm{2}} \:+\:{x}_{\mathrm{3}} \:+\:{x}_{\mathrm{4}} \:=\:\mathrm{5} \\ $$$$\mathrm{with} \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{1}} \:\leqslant\:\mathrm{3}\:\: \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{2}} \:\leqslant\:\mathrm{3} \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{3}} \:\leqslant\:\mathrm{2} \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{4}} \:\leqslant\:\mathrm{2} \\ $$

Commented by 121194 last updated on 26/Mar/19

3 2 0 0 3 1 1 0 3 1 0 1 3 0 2 0 3 0 1 1 3 0 0 2 2 3 0 0 2 2 1 0 2 2 0 1 2 1 2 0 2 1 1 1 2 1 0 2 2 0 2 1 2 0 1 2 1 3 1 0 1 3 0 1 1 2 2 0 1 2 1 1 1 2 0 2 1 1 2 1 1 1 1 2 1 0 2 2 0 3 2 0 0 3 1 1 0 3 0 2 0 2 2 1 0 2 1 2 0 1 2 2 28

$$\mathrm{3}\:\mathrm{2}\:\mathrm{0}\:\mathrm{0} \\ $$$$\mathrm{3}\:\mathrm{1}\:\mathrm{1}\:\mathrm{0} \\ $$$$\mathrm{3}\:\mathrm{1}\:\mathrm{0}\:\mathrm{1} \\ $$$$\mathrm{3}\:\mathrm{0}\:\mathrm{2}\:\mathrm{0} \\ $$$$\mathrm{3}\:\mathrm{0}\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{3}\:\mathrm{0}\:\mathrm{0}\:\mathrm{2} \\ $$$$\mathrm{2}\:\mathrm{3}\:\mathrm{0}\:\mathrm{0} \\ $$$$\mathrm{2}\:\mathrm{2}\:\mathrm{1}\:\mathrm{0} \\ $$$$\mathrm{2}\:\mathrm{2}\:\mathrm{0}\:\mathrm{1} \\ $$$$\mathrm{2}\:\mathrm{1}\:\mathrm{2}\:\mathrm{0} \\ $$$$\mathrm{2}\:\mathrm{1}\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{2}\:\mathrm{1}\:\mathrm{0}\:\mathrm{2} \\ $$$$\mathrm{2}\:\mathrm{0}\:\mathrm{2}\:\mathrm{1} \\ $$$$\mathrm{2}\:\mathrm{0}\:\mathrm{1}\:\mathrm{2} \\ $$$$\mathrm{1}\:\mathrm{3}\:\mathrm{1}\:\mathrm{0} \\ $$$$\mathrm{1}\:\mathrm{3}\:\mathrm{0}\:\mathrm{1} \\ $$$$\mathrm{1}\:\mathrm{2}\:\mathrm{2}\:\mathrm{0} \\ $$$$\mathrm{1}\:\mathrm{2}\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{1}\:\mathrm{2}\:\mathrm{0}\:\mathrm{2} \\ $$$$\mathrm{1}\:\mathrm{1}\:\mathrm{2}\:\mathrm{1} \\ $$$$\mathrm{1}\:\mathrm{1}\:\mathrm{1}\:\mathrm{2} \\ $$$$\mathrm{1}\:\mathrm{0}\:\mathrm{2}\:\mathrm{2} \\ $$$$\mathrm{0}\:\mathrm{3}\:\mathrm{2}\:\mathrm{0} \\ $$$$\mathrm{0}\:\mathrm{3}\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{0}\:\mathrm{3}\:\mathrm{0}\:\mathrm{2} \\ $$$$\mathrm{0}\:\mathrm{2}\:\mathrm{2}\:\mathrm{1} \\ $$$$\mathrm{0}\:\mathrm{2}\:\mathrm{1}\:\mathrm{2} \\ $$$$\mathrm{0}\:\mathrm{1}\:\mathrm{2}\:\mathrm{2} \\ $$$$\mathrm{28} \\ $$

Commented by Joel578 last updated on 27/Mar/19

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

Answered by mr W last updated on 26/Mar/19

using generating functions x_1 : 1+t+t^2 +t^3 x_2 : 1+t+t^2 +t^3 x_3 : 1+t+t^2 x_4 : 1+t+t^2 x_1 +x_2 +x_3 +x_4 : (1+t+t^2 +t^3 )^2 (1+t+t^2 )^2 =(((1−t^4 )^2 (1−t^3 )^2 )/((1−t)^4 )) =Σ_(k=0) ^2 C_k ^2 (−1)^k t^(4k) Σ_(k=0) ^2 C_k ^2 (−1)^k t^(3k) Σ_(k=0) ^∞ C_k ^(3+k) t^k coef. of term t^5 : C_1 ^2 (−1)^1 C_0 ^2 (−1)^0 C_1 ^4 +C_0 ^2 (−1)^0 C_1 ^2 (−1)^1 C_2 ^5 +C_0 ^2 (−1)^0 C_0 ^2 (−1)^0 C_5 ^8 =−C_1 ^2 C_0 ^2 C_1 ^4 −C_0 ^2 C_1 ^2 C_2 ^5 +C_0 ^2 C_0 ^2 C_5 ^8 =−2×4−2×10+56 =28 ⇒there are 28 possible solutions for x_1 + x_2 + x_3 + x_4 = 5 under given conditions. other terms see below:

$${using}\:{generating}\:{functions} \\ $$$${x}_{\mathrm{1}} :\:\mathrm{1}+{t}+{t}^{\mathrm{2}} +{t}^{\mathrm{3}} \\ $$$${x}_{\mathrm{2}} :\:\mathrm{1}+{t}+{t}^{\mathrm{2}} +{t}^{\mathrm{3}} \\ $$$${x}_{\mathrm{3}} :\:\mathrm{1}+{t}+{t}^{\mathrm{2}} \\ $$$${x}_{\mathrm{4}} :\:\mathrm{1}+{t}+{t}^{\mathrm{2}} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} :\:\left(\mathrm{1}+{t}+{t}^{\mathrm{2}} +{t}^{\mathrm{3}} \right)^{\mathrm{2}} \left(\mathrm{1}+{t}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} =\frac{\left(\mathrm{1}−{t}^{\mathrm{4}} \right)^{\mathrm{2}} \left(\mathrm{1}−{t}^{\mathrm{3}} \right)^{\mathrm{2}} }{\left(\mathrm{1}−{t}\right)^{\mathrm{4}} } \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}} {\sum}}{C}_{{k}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{{k}} {t}^{\mathrm{4}{k}} \underset{{k}=\mathrm{0}} {\overset{\mathrm{2}} {\sum}}{C}_{{k}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{{k}} {t}^{\mathrm{3}{k}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{{k}} ^{\mathrm{3}+{k}} {t}^{{k}} \\ $$$$ \\ $$$${coef}.\:{of}\:{term}\:{t}^{\mathrm{5}} : \\ $$$${C}_{\mathrm{1}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{1}} {C}_{\mathrm{0}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{0}} {C}_{\mathrm{1}} ^{\mathrm{4}} +{C}_{\mathrm{0}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{0}} {C}_{\mathrm{1}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{1}} {C}_{\mathrm{2}} ^{\mathrm{5}} +{C}_{\mathrm{0}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{0}} {C}_{\mathrm{0}} ^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{0}} {C}_{\mathrm{5}} ^{\mathrm{8}} \\ $$$$=−{C}_{\mathrm{1}} ^{\mathrm{2}} {C}_{\mathrm{0}} ^{\mathrm{2}} {C}_{\mathrm{1}} ^{\mathrm{4}} −{C}_{\mathrm{0}} ^{\mathrm{2}} {C}_{\mathrm{1}} ^{\mathrm{2}} {C}_{\mathrm{2}} ^{\mathrm{5}} +{C}_{\mathrm{0}} ^{\mathrm{2}} {C}_{\mathrm{0}} ^{\mathrm{2}} {C}_{\mathrm{5}} ^{\mathrm{8}} \\ $$$$=−\mathrm{2}×\mathrm{4}−\mathrm{2}×\mathrm{10}+\mathrm{56} \\ $$$$=\mathrm{28} \\ $$$$ \\ $$$$\Rightarrow{there}\:{are}\:\mathrm{28}\:{possible}\:{solutions}\:{for} \\ $$$${x}_{\mathrm{1}} \:+\:{x}_{\mathrm{2}} \:+\:{x}_{\mathrm{3}} \:+\:{x}_{\mathrm{4}} \:=\:\mathrm{5} \\ $$$${under}\:{given}\:{conditions}. \\ $$$$ \\ $$$${other}\:{terms}\:{see}\:{below}: \\ $$

Commented by mr W last updated on 26/Mar/19

Commented by mr W last updated on 26/Mar/19

x_1 +x_2 +x_3 +x_4 =0 ⇒ 1 solution x_1 +x_2 +x_3 +x_4 =1 ⇒ 4 solutions x_1 +x_2 +x_3 +x_4 =2 ⇒ 10 solutions x_1 +x_2 +x_3 +x_4 =3 ⇒ 18 solutions x_1 +x_2 +x_3 +x_4 =4 ⇒ 25 solutions x_1 +x_2 +x_3 +x_4 =5 ⇒ 28 solutions x_1 +x_2 +x_3 +x_4 =6 ⇒ 25 solutions x_1 +x_2 +x_3 +x_4 =7 ⇒ 18 solutions x_1 +x_2 +x_3 +x_4 =8 ⇒ 10 solutions x_1 +x_2 +x_3 +x_4 =9 ⇒ 4 solutions x_1 +x_2 +x_3 +x_4 =10 ⇒ 1 solution

$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{0}\:\Rightarrow\:\mathrm{1}\:{solution} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{1}\:\Rightarrow\:\mathrm{4}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{2}\:\Rightarrow\:\mathrm{10}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{3}\:\Rightarrow\:\mathrm{18}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{4}\:\Rightarrow\:\mathrm{25}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{5}\:\Rightarrow\:\mathrm{28}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{6}\:\Rightarrow\:\mathrm{25}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{7}\:\Rightarrow\:\mathrm{18}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{8}\:\Rightarrow\:\mathrm{10}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{9}\:\Rightarrow\:\mathrm{4}\:{solutions} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} =\mathrm{10}\:\Rightarrow\:\mathrm{1}\:{solution} \\ $$

Commented by Joel578 last updated on 27/Mar/19

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

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