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Question Number 173992 by dragan91 last updated on 22/Jul/22

Commented by mr W last updated on 22/Jul/22

$$noanswerseemstobecorrect.igot$$$$5456.$$

Answered by mr W last updated on 22/Jul/22

$$x_1+x_2+x_3=nhas{C}_2^{n+2}solutions$$$$with{x}_i⩾0.$$$$totalnumberofsolutionsfor0⩽n⩽30:$$$$\underset{n=0}{\sum ^{30}}{C}_2^{n+2}$$$$=\underset{n=0}{\sum ^{30}}\frac{(n+2)(n+1)}{2}$$$$=\underset{n=0}{\sum ^{30}}\frac{(n^2+3n+2)}{2}$$$$=\frac{1}{2}(\frac{30\times 31\times 61}{6}+3\times \frac{30\times 31}{2}+31\times 2)$$$$=5456$$

Commented by dragan91 last updated on 22/Jul/22

$$\mathrm{n}\mathrm{starts}\mathrm{from}3\mathrm{since}{\mathrm{x}}_i\in \mathrm{N}$$

Commented by dragan91 last updated on 22/Jul/22

$$\mathrm{Number}\mathrm{of}\mathrm{positive}\mathrm{integers}\mathrm{if}\mathrm{equation}$$$${\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3=\mathrm{n}\mathrm{for}\mathrm{n}⩾3\mathrm{is}{\mathrm{C}}_2^{\mathrm{n}-1}$$$$$$

Commented by dragan91 last updated on 22/Jul/22

$$\mathrm{At}\mathrm{first}\mathrm{I}\mathrm{get}4060$$

Commented by mr W last updated on 22/Jul/22

$$tome0\in \mathbb{N}.besidesthequestionstates$$$$0⩽x_1+x_2+x_3⩽30.$$

Commented by mr W last updated on 22/Jul/22

Commented by dragan91 last updated on 22/Jul/22

$$\mathrm{Ok}.\mathrm{For}\mathrm{me}0\mathrm{is}\mathrm{not}\mathrm{natural}.\mathrm{I}\mathrm{will}\mathrm{check}\mathrm{with}$$$$\mathrm{other}\mathrm{people}.\mathrm{I}\mathrm{think}\mathrm{the}\mathrm{given}\mathrm{answers}\mathrm{is}\mathrm{all}\mathrm{wrong}.\mathrm{Thanks}.$$$$$$

Commented by Tawa11 last updated on 22/Jul/22

$$\mathrm{Great}\mathrm{sir}$$

Commented by mr W last updated on 22/Jul/22

$$itisjustaquestionofdefinition$$$$if0isanaturalnumberornot.$$$$inGermany{it}^{\prime }sevendifferently$$$$definedindifferentfederalstates.$$

Commented by dragan91 last updated on 22/Jul/22

$$\mathrm{thanks}$$

Commented by MJS_new last updated on 23/Jul/22

$$\mathrm{once}\mathrm{we}\mathrm{used}$$$$\mathbb{N}=\{1,2,3,...\}\mathrm{and}{\mathbb{N}}_0=\{0,1,2,3,...\}$$$$\mathrm{now}\mathrm{most}\mathrm{people}\mathrm{use}$$$$\mathbb{N}=\{0,1,2,3,...\}\mathrm{and}{\mathbb{N}}^{★}=\{1,2,3,...\}$$

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