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-

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

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

Commented by Joel578 last updated on 27/Mar/19

t h a n k y o u v e r y m u c h

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

u s i n g g e n e r a t i n g f u n c t i o n s

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} = \frac{{(1 - t^{4})}^{2} {(1 - t^{3})}^{2}}{{(1 - t)}^{4}}

= \underset{k = 0}{\sum^{2}} C_{k}^{2} {(- 1)}^{k} t^{4 k} \underset{k = 0}{\sum^{2}} C_{k}^{2} {(- 1)}^{k} t^{3 k} \underset{k = 0}{\sum^{\infty}} C_{k}^{3 + k} t^{k}

c o e f . o f t e r m 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 \times 4 - 2 \times 10 + 56

= 28

\Rightarrow t h e r e a r e 28 p o s s i b l e s o l u t i o n s f o r

x_{1} + x_{2} + x_{3} + x_{4} = 5

u n d e r g i v e n c o n d i t i o n s .

o t h e r t e r m s s e e b e l o w :

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 \Rightarrow 1 s o l u t i o n

x_{1} + x_{2} + x_{3} + x_{4} = 1 \Rightarrow 4 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 2 \Rightarrow 10 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 3 \Rightarrow 18 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 4 \Rightarrow 25 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 5 \Rightarrow 28 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 6 \Rightarrow 25 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 7 \Rightarrow 18 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 8 \Rightarrow 10 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 9 \Rightarrow 4 s o l u t i o n s

x_{1} + x_{2} + x_{3} + x_{4} = 10 \Rightarrow 1 s o l u t i o n

Commented by Joel578 last updated on 27/Mar/19

t h a n k y o u v e r y m u c h

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