5 votes 5 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 ? Combinatory discrete-mathematics kenneth-rosen combinatory + – Nirmal Gaur asked Apr 14, 2017 edited Mar 5, 2019 by Pooja Khatri Nirmal Gaur 9.7k views answer comment Share Follow See all 10 Comments See all 10 10 Comments reply Show 7 previous comments Pinaki Dash commented Sep 1, 2017 reply Follow Share Same as this: https://gateoverflow.in/132586/discrete-maths-counting-number-of-solutions-to-the-equation 1 votes 1 votes tirth_patel commented Jul 4, 2021 reply Follow Share Best explanation! https://www.slader.com/discussion/question/how-many-solutions-are-there-to-the-equation-x_1-x_2-x_3-x_4-x_5-21/ 0 votes 0 votes Arjun commented Jul 4, 2021 reply Follow Share @asqwer Please do not spam with links 0 votes 0 votes Please log in or register to add a comment.
1 votes 1 votes $\begin{align*} &\Rightarrow x_1 + x_2 + x_3 + x_4 + x_5 = 21 \quad \left ( 0\leq x_i \leq 10 \right ) \\ \\ &\Rightarrow \text{No of integral solution of the above equation is } \\ &\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left ( 1+x+x^2+ \dots + x^{10} \right )^5 \\ &\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left [ \frac{1-x^{11}}{1-x} \right ]^5 \\ &\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left [ 1-x^{11} \right ]^5 \cdot \left [ \frac{1}{\left ( 1-x \right )^5} \right ] \\ &\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \left [ \sum_{m=0}^{5}\binom{5}{m}\left ( -1 \right )^m \cdot x^{11\cdot m} \right ] \cdot \left [ \sum_{r=0}^{\infty}\binom{5+r-1}{r}\cdot x^r \right ] \\ &\Rightarrow {\color{red}{\left [ x^{21} \right ]}} \begin{cases} &+1 \cdot \binom{25}{21} \qquad m = 0,r = 21 \\ \\ &-5 \cdot \binom{14}{10} \qquad m = 1,r = 10 \\ \end{cases} \\ &\Rightarrow {\color{red}{\left [ x^{21} \right ]}} = +1 \cdot \binom{25}{21} -5 \cdot \binom{14}{10} = {\bf \color{red}{7645}} \\ \end{align*}$ more examples dd answered Apr 14, 2017 dd comment Share Follow See 1 comment See all 1 1 comment reply Akriti sood commented Apr 15, 2017 reply Follow Share but constraint is for x1 that it lies btw 0 and 10.why r u taking it for all?? 0 votes 0 votes Please log in or register to add a comment.