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


Question and Answers Forum

All Questions Topic List
Permutation and Combination Questions
Previous in All Question Next in All Question
Previous in Permutation and Combination Next in Permutation and Combination
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

	$t h a n k y o u v e r y m u c h$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum