4 votes

How many solutions are there to the equation

x1 + x2 + x3 + x4 + x5 = 21,

where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: 0$\leq$ x1$\leq$10 ?

x1 + x2 + x3 + x4 + x5 = 21,

where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: 0$\leq$ x1$\leq$10 ?

0

@ Nirmal Gaur . please check bound condition. is it correct ? or, just $x_1$

$\left ( 0\leq x_i \leq 10 \right )$

5 votes

Best answer

x1 + x2 + x3 + x4 + x5 = 21, where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: **0≤ x1 ≤10 **and x2,x3,x4,x5 can range in between 0 to 21.

Now we have to choose the coefficient of x^{21 }in following expression

(1+ x + x^{2} +x^{3}.....+ x^{10}) (1+ x + x^{2} +x^{3}.......+X^{21})(1+ x + x^{2} +x^{3}.....+ x^{21})(1+ x + x^{2} +x^{3}.....+ x^{21})(1+ x + x^{2}+x^{3}.....+ x^{21})

also we can extend term of x2,x3,x4,x5 from x^{21 }upto infinty because it didn't change the value of x^{21} coefficient .

(1+ x + x^{2} +x^{3}.....+ x^{10}) (1+ x + x^{2} +x^{3}.......**∞**)(1+ x + x^{2} +x^{3}.....+ **∞**)(1+ x + x^{2} +x^{3}.....+ **∞**)(1+ x + x^{2}+x^{3}.....+ **∞**)

$\left ( \frac{1-x^{11}}{1-x} \right )\left ( \frac{1}{(1-x)^{4}} \right )$

$\left ( 1-x^{11} \right )\left ( \frac{1}{(1-x)^{5}} \right )$

now we have to find only coefficient of x^{21 }for which we consider coefficient of x^{21} and x^{10 }in $\left ( 1-x \right )^{-5}$

general term of binomial coefficient is $\binom{n+r-1}{r}$

$\binom{5+21-1}{21}-\binom{5+10-1}{10}$=**12650-1001**=**11649**

6 votes

x1 + x2 + x3 + x4 + x5 = 21, where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: **0≤ x1 ≤10 .**

Which can be solved by formula : $\binom{25}{4} - \binom{14}{4}$

Total possibilites without upper Bound : $\binom{25}{4}$

possibilites with upper Bound: $\binom{14}{4}$

Answer will be 11,649.

6 votes

x1 + x2 + x3 + x4 + x5 = 21

0≤x1 ≤3 ,1≤ x2<4 and x3≥15

case 1: x3≥15 given to make condition <x1, x2, x3, x4, x5> ≥ 0 remove 15 from 21.

so new equation becomes x1 + x2 + x3 + x4 + x5 = 21-15 =6

case 2: 1≤ x2<4=1≤ x2≤3 given so make it 0≤ x2 ≤2 for that remove 1 from 6.

so new equation becomes x1 + x2 + x3 + x4 + x5 =6-1=5

find all solutions where <x1, x2, x3, x4, x5> ≥ 0 lets say it solution A.

and remove the solution where 0≤ x2 ≤2 that is we will remove all solutions from A where x2>2 and all other variables are xi>=0 let's say it solution B.

0≤x1 ≤3 that is we will remove all solution from A where x1>3 and all other variables are xi>=0 let's say it solution C.

^{A= (5+5-1)}C_{ (5-1) }

= 126

C= 5C4 =5 solutions

B=6C4 =15 solutions

B and C =0 solutions

Answer = A - (B+C - B and C)

=126-20

=106

3 votes

x1 + x2 + x3 + x4 + x5 = 21, where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: **0≤ x1 ≤10 ,0≤ x2 ≤10 ,0≤ x3 ≤10 ,0≤ x4 ≤10 ,0≤ x5 ≤10 ,**

Solved by Generating Function :

(1+ x + x^{2} +x^{3}.....+ x^{10}) (1+ x + x^{2} +x^{3}.....+ x^{10})(1+ x + x^{2} +x^{3}.....+ x^{10})(1+ x + x^{2} +x^{3}.....+ x^{10})(1+ x + x^{2} +x^{3}.....+ x^{10}) = x^{21}.

((1+ x + x^{2} +x^{3}.....+ x^{10}))^{5} = x^{21}.

$\left ( \frac{1- x^{11}}{1-x} \right )^{5}$ = x^{21}.

$\left ( 1- x^{11} \right )^{5}\left ( \frac{1}{1-x} \right )^{5}$ = x^{21}.

$\left ( 1- x^{11} \right )^{5}\sum_{r=0}^{n}\binom{n+r-1}{r} x^{r} = x^{21}.$

$\left ( 1- x^{11} \right )^{5}\sum_{r=0}^{n}\binom{5+r-1}{r} x^{r} = x^{21}.$

$\left ( 1- x^{11} \right )^{5}\sum_{r=0}^{n}\binom{r+4}{r} x^{r} = x^{21}.$

Now we have to find term of x whose cofficient is 21.

$\left (\binom{5}{0} x^{0} - \binom{5}{1}x^{11} + \binom{5}{2}x^{22} - \binom{5}{3}x^{33}+ \binom{5}{4}x^{44} - \binom{5}{5}x^{55} \right )$ $\times \sum_{r=0}^{n}\binom{r+4}{r} x^{r} = x^{21}.$

$\left ( _{0}^{5}\textrm{C} \times _{21}^{25}\textrm{C}x^{21} \right ) - \left ( _{1}^{5}\textrm{C} \times _{10}^{14}\textrm{C}x^{21} \right ) = x^{21}$

$x^{21}\left ( 1 \times \left ( 25\times 23\times 22 \right ) \right ) - \left ( 5 \times\left ( 7\times 13\times 11 \right ) \right ) = x^{21}$

= 7645

1 vote

&\Rightarrow x_1 + x_2 + x_3 + x_4 + x_5 = 21 \quad \left ( 0\leq x_i \leq 10 \right ) \\ \\

&\Rightarrow \text{No of integral solution of the above equation is } \\

&\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left ( 1+x+x^2+ \dots + x^{10} \right )^5 \\

&\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left [ \frac{1-x^{11}}{1-x} \right ]^5 \\

&\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left [ 1-x^{11} \right ]^5 \cdot \left [ \frac{1}{\left ( 1-x \right )^5} \right ] \\

&\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left [ \sum_{m=0}^{5}\binom{5}{m}\left ( -1 \right )^m \cdot x^{11\cdot m} \right ] \cdot \left [ \sum_{r=0}^{\infty}\binom{5+r-1}{r}\cdot x^r \right ] \\

&\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \begin{cases}

&+1 \cdot \binom{25}{21} \qquad m = 0,r = 21 \\ \\

&-5 \cdot \binom{14}{10} \qquad m = 1,r = 10 \\

\end{cases} \\

&\Rightarrow {\color{red}{\left [ x^{21} \right ]}} = +1 \cdot \binom{25}{21} -5 \cdot \binom{14}{10} = {\bf \color{red}{7645}} \\

\end{align*}$